Infinity is out of domain of integer numbers where notions of even and odd are defined and make sense.
Notion of infinity is applicable when we are discussing sequences and their behaviour, such as convergence.
Convergence of sequence x(n) to infinity by definition is: for each real number ε>0 there exists a natural number N(ε) such that for every number n≥N(ε) we have |x(n)|>ε.
Convergence of sequence x(n) to plus infinity by definition is: for each real number ε>0 there exists a natural number N(ε) such that for every number n≥N(ε) we have x(n)>ε.
Convergence of sequence x(n) to minus infinity by definition is: for each real number ε>0 there exists a natural number N(ε) such that for every number n≥N(ε) we have x(n)For example sequence of natural numbers 1, 2, 3... converges to plus infinity and to infinity; sequence of negated natural numbers -1, -2, -3... converges to minus infinity and to infinity; and sequence of sign-alternating numbers (-1)^n * n: -1, 2, -3, 4, -5, 6, -7, 8, -9, 10... converges to infinity.
So notion of infinity applies to behaviour of sequences, whose elements remain finite nevertheless. If we consider other mathematical objects, e.g. integer numbers, then notion of infinity does not apply. If we consider convergence of sequences where notion of infinity is applicable, then notion of even/odd is not applicable.
While discussing sequences converging to an infinity with a child, it may be useful to consider some interesting counterexamples: sequences which are unbounded, but still do not converge to infinity, e.g. 1, 2, 1, 4, 1, 6, 1, 8, 1, 10, 1, 12, 1, 14, 1, 16, 1, 18, 1, 20... (formula is n^((1+(-1)^n)/2)).