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Traditionally,with each other is used for two things, and with one another is used for more. But as the website I link to says, this distinction is disappearing. (Although to me, with one another still doesn't sound quite right for two things.)
I'm not saying the estimated prevalence from Google Books is completely accurate, but it claims 21,800 hits for they are on a par with, against only 1,520 for they are on par with. Personally, I'd have expected a more marked preference, but even at that level your assertion is clearly incorrect so far as actual usage is concerned (and I never heard of any pedants supporting your position on logical / syntactic grounds).
(of two or more things, with reference to two or more things previously mentioned) referring or applying to in a parallel or sequential way: Joe and Bob escorted Betty and Alice, respectively.
+1. My engineering masters had many instances of this, not matter how hard I tried to mix the writing up so as not to repeat myself I ended up reverting all the other clumsy attempts back to variations of 'respectively'.
@Vector Apologies. But I honestly didn't see your answer, and am quite mystified as to why mine has scored three positive votes, and yours a negative two - especially since you provided more information. Anyway +1 from me.
@DarrenRinger also used parity in another answer, below. IMO it works much better than the answers here that are fumbling around to make in par work. I added the information about usage of parity later - I think it was down-voted before because it was too short not everyone understood how parity can be used.
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.
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