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If two integers are either both negative or both positive, we can say they have the same sign.

How about two integers that are either both odd or both even? Is there any term for them?

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    $\begingroup$ en.wikipedia.org/wiki/Parity_(mathematics) $\endgroup$
    – TomKern
    Commented Apr 23, 2023 at 3:18
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    $\begingroup$ You can also say they are congruent mod 2. $\endgroup$ Commented Apr 23, 2023 at 3:22
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    $\begingroup$ As @TomKern says: they have the same parity. If one is odd and the other even, then they have opposite parity. $\endgroup$ Commented Apr 23, 2023 at 9:01

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Expanding on a comment by @TomKern: The word you're looking for is parity.

You can see it in most standard English dictionaries, such as Merriam-Webster, here:

4 a: the property of an integer with respect to being odd or even

3 and 7 have the same parity

b(1): the state of being odd or even used as the basis of a method of detecting errors in binary-coded data (see binary entry 2 sense 5)

b(2): parity bit

And here's the Wikipedia article on the subject.

In mathematics, parity is the property of an integer of whether it is even or odd.

And here's the equivalent question at Stack Exchange: English Language & Usage.

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