1:equality, as in amount, status, or character. 2:equivalence;
correspondence; similarity; analogy.
British Dictionary definitions for parity:
1:equality of rank, pay, etc 2:close or exact analogy or equivalence
Parity is a noun and is generally used to represent a state of equality/equivalence, etc. Thus, 'in parity' suggests that each group, the A's, and the B's, is in a state of parity with respect to itself - a state of equality or equilibrium (not the same), which appears to be what you want to express.