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I checked the definition of parity here. It is defined as:

1 The state or condition of being equal, especially as regards status or pay.

Then I found following example on the same site:

That is far from the truth as I also want a better world with equal parity to men for my wife, daughters, sisters and so on.

As per definition above, parity itself means state being equal.

Then why word equal is additionally included in above example sentence?

It appears to be a redundant word.

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    If you leaf through some of about 235 results in Google Books for with equal parity, you'll see that most of them are for either the maths sense (OED: the property of an integer by virtue of which it is odd or even), or the physics sense (OED: the property of having or being a spatial wave function that either remains the same (has even parity) or changes sign (has odd parity) when a change of sign is applied to the coordinates). Unlike those, your cited example is indeed tautologous, and sounds a bit clumsy to me. – FumbleFingers Jun 12 '17 at 14:41
  • ...having said that, since we usually talk about women having parity with men, that creates its own problem in the context of wanting to live in a world with that characteristic. – FumbleFingers Jun 12 '17 at 14:43
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It sounds like an awkwardly-constructed sentence to me. People normally say, "for women to have parity with men" or "for their to be parity between men and women". That is, we don't normally say "parity to" but "parity with". I think the writer wanted to put "men" first so that he could end the sentence with the string "wife, daughters, sisters, and so on", to give the emphasis and rhythm that he wanted. And then maybe " a world with parity with" sounded odd because of the two "with"s so close together. And yes, "equal parity" in this context is redundant. As opposed to what, unequal parity?

Note that the word "parity" has other meanings. For example, to mathematicians and computer people, "parity" refers to whether a number is odd or even. That is, someone might ask, "What is the parity of this number?", and reasonable answers would be "odd" or "even". Or you can say a number has "even parity" or "odd parity". In that case it would make sense to say, "These two numbers have equal parity", meaning that they are either both odd or they are both even.

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The two parities implied here are the two relationships men->women, and women->men.

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