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.