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By $MRDP$ resolution of Hilbert's tenth, we infer, counting number of solutions to Diophantine equations is undecidable.

Is parity of number of solutions to Diophantine equations undecidable?

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While it’s unclear what you mean by “parity” when the number of solutions is infinite, all such questions have been answered by M. Davis, On the number of solutions of Diophantine equations, Proc. AMS 35 (1972), 552–554, doi link:

Theorem: Fix $\varnothing\subsetneq A\subsetneq\omega\cup\{\aleph_0\}$. Then it is undecidable whether the number of solutions of a given Diophantine equation belongs to $A$.

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