The parity of a sum or product is fixed before you compute it

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What to assume they know

Odd and even as "can this be arranged in pairs": even as a count that lays out in twos, odd as that picture with one dot over, and the parity of a sum of several numbers
Multiplication read as repeated addition, and as counting a rectangular array
Brackets decide which operation happens first and the Class 6 work on letter-numbers: a letter standing for a position in a sequence
Reading a table of values for an expression
Multiples of 2 and of 4, and the idea of a factor

What they should be able to do

Complete the parity table for addition and for subtraction, and justify each row from the dot picture rather than from examples
State the parity of a sum from nothing but how many odd numbers it contains
Say whether an m × n grid holds an odd or an even count of unit squares, from m and n alone and without multiplying
Explain why one even side is enough to make a rectangular count even
Evaluate an expression at several values of the letter-number and read its parity off the table
Sort expressions into those with a fixed parity for every value and those whose parity changes, and say which feature of the expression decides that
Write the nth even number and the nth odd number as expressions, and say which numbers each expression reaches
Judge a claim of the form "this expression always gives odd numbers" as a claim about every value, and refute it with one counterexample where it is false

Where it usually goes wrong

"Subtraction will need its own set of rules." It will not. Taking away a block of pairs never touches a leftover, so subtraction copies addition row for row. Deriving that, instead of memorising four more lines, is section 2.
"An odd number times an even number could be odd." Never: an even factor means the whole array pairs up along one direction. The grid picture makes this visible in a way the times table does not — one even side, and the cells fall into columns of two.
"3n + 4 is even, because 4 is even." It is even only when 3n is. The coefficient 3 is odd, so 3n takes the parity of n and the expression flips with it. The printed table is built to catch exactly this.
"An expression that always gives even values gives all the even values." 6k + 2 is the counterexample the chapter prints. Always-even is a statement about the outputs it produces; listing all the evens is a much stronger claim about the ones it does not miss.
"2p + 1 and 2q − 1 are the same thing, so both list every odd number." As expressions they differ only by where the count starts, and in this chapter a letter-number counts positions beginning at one. Under that convention 2q − 1 starts at 1 while 2p + 1 starts at 3 and never reaches 1. Say the convention out loud; without it the claim is neither true nor false.
"Parity checks out, so the answer is yes." Matching parity only means the parity argument has nothing to say. The loose-sheets question on p.144 is exactly this trap — see Notes, where the finer argument is set out.
"77 is odd, so the bulb ends up odd." A bulb is not a number. What the parity of 77 controls is whether the state has been flipped an even or an odd number of times, and only that.

Questions to check understanding

Complete a parity table across addition and subtraction and justify one row
Give the parity of a product from its factors' parities, with a reason
Say whether an m × n grid holds an odd or even number of cells, without multiplying
Tabulate an expression at several values and state whether its parity is fixed
Produce an expression with a stated parity behaviour, and one that changes
Identify which of several parity claims about expressions are true, and give a counterexample for each false one (the p.144 set is exactly this)
Fill a small grid so that every row sum and every column sum has a stated parity, using a stated number of odd and even entries (question 3 on p.144)
Toggle-and-state questions, and "can this total be reached" questions, both of which reduce to counting how many odd contributions there are
Give the nth even and nth odd number, and use them to find a term at a stated position

Examples worth working on the board

The parity table (Part I, §6.2, p.131). Three rows are printed complete — even with even, odd with odd, even with odd — and four subtraction rows are left blank.
Lakpa's piggy bank (Part I, §6.2, p.131). Inputs: ₹1 coins in an odd count, ₹5 coins also in an odd count, ₹10 coins in an even count, and ₹205 claimed as the total. Do not pre-compute the verdict.
The grid figure (Part I, "Small Squares in Grids", §6.2, p.131). Checked against the printed page: a single blank grid printed in the margin, ruled three rows by four columns, twelve cells, no numerals inside it. The text beside it contrasts a 3 × 3 grid holding 9 small squares with a 3 × 4 grid holding 12.
Three grids to call (Part I, §6.2, p.132). Inputs, exactly as printed: 27 × 13, 42 × 78 and 135 × 654. The instruction attached to them is the point — say the parity without doing the multiplication. Two of the three have at least one even side; that is all the working there is.
The 3n + 4 table (Part I, "Parity of Expressions", §6.2, p.132). Checked against the printed page. Three columns headed for n, for the value, and for the parity of the value, with three rows: n = 3 giving 13, n = 8 giving 28, n = 10 giving 34.
Printed always-even examples (Part I, §6.2, p.132). 100p and 48w − 2. Both have an even coefficient, which is exactly why the letter cannot affect the parity. The chapter then asks for always-odd examples and for further parity-changing ones; those it leaves open.
6k + 2 (Part I, §6.2, p.132). Printed with its first values 8, 14, 20 for k = 1, 2, 3. Every value is even, and yet most even numbers never appear. This is the example that separates "always even" from "gives all the evens".
The 100th even and the 100th odd (Part I, §6.2, pp.132–133). Inputs: the two sequences written under one another, and the observation that at each position the odd one sits one below the even one. The chapter arrives at 2n and 2n − 1 from this comparison, and it is worth showing the comparison rather than asserting the formulas.
The four claims (Part I, §6.5 Figure it Out, p.144, question 10). Inputs: 4m − 1 always gives odd numbers; every even number can be written as 6j − 4; 2p + 1 and 2q − 1 both list all the odd numbers; 2f + 3 gives both parities. Exactly one is true.
The light bulb (Part I, §6.5 Figure it Out, p.143, question 1). A bulb that starts on and is switched 77 times. Parity of the count of switchings decides the final state; nothing else about 77 matters. The Figure it Out heading and this first question sit at the foot of p.143, below the four side cryptarithms; the rest of the set runs over onto p.144.
The parity grid (Part I, §6.5 Figure it Out, p.144, question 3). Checked against the printed page: a 2 × 3 grid of empty cells with a small coloured circle against each row and each column carrying the parity that line's sum must have — the two rows marked odd then even, the three columns marked even, even, odd. Six numbers go in, three of them odd and three even.

Figures to have open

A rectangular grid whose two dimensions can be set independently and whose cells can be swept into pairs. This is the load-bearing figure of the topic — the product argument is not convincing without it.
A three-column table for an expression, its value and the parity of the value, fillable row by row.
A number strip of the even numbers with selected entries highlighted, for the 6k + 2 gap.
The even and odd sequences aligned vertically, position by position.
No photograph, dataset or printed illustration from the textbook is required; the printed grid on p.131 is blank and can simply be redrawn.

Where this sits in the book

NCERT Ganita Prakash, Class 7, Part I, printed Chapter 6 "Number Play", §6.2 "Picking Parity", pp.131–133 — the Figure it Out parity set and the piggy-bank question (p.131), the unnumbered block "Small Squares in Grids" (pp.131–132), the unnumbered block "Parity of Expressions" with the 3n + 4 table (p.132), and the comparison of the even and odd sequences ending in 2n and 2n − 1 (pp.132–133)
Same part, §6.5 Figure it Out, pp.143–144, questions 1, 3, 5, 6 and 10, which are parity questions collected at the end of the chapter rather than at the end of §6.2. The heading and question 1 are printed on p.143; questions 3, 5, 6 and 10 are on p.144, as is question 2, which is discussed in Notes
Same part, SUMMARY, p.145, third bullet, which is the chapter's own claim to have covered products as well as sums
Solutions appendix bound after p.145 in the cached PDF (not part of the printed book), consulted only to confirm the intended verdicts on the p.132 questions and on the Figure it Out set that runs from p.143 onto p.144