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In the context of transfinite ordinals, the usual definition is that an ordinal number $\alpha$ is even if it is a multiple of $2$, specifically: if there is another ordinal $\beta$ such that $2\cdot\beta=\alpha$. In other words, the order type $\alpha$ can be viewed as $\beta$ many pairs in sequence, or in other words, $\alpha$ is left-divisible by $2$. Otherwise, it is odd.

It is easy to prove from this definition by transfinite recursion that the ordinals come in an alternating even/odd pattern, and that every limit ordinal (and hence every infinite cardinal) is even. Many transfinite constructions proceed by doing something different on the even as opposed to the odd stages, just as with finite constructions.

The smallest infinite ordinal is $\omega$, which is even on this definition, since having $\omega$ many pairs in sequence is order-isomorphic to $\omega$, and so $2\cdot\omega=\omega$. Meanwhile, the next infinite ordinal is $\omega+1$, which is odd. The ordinal $\omega+2$ is even, since it is equal to $2\cdot(\omega+1)$, even though it is not $\beta+\beta$ for any $\beta$.

(Please note that $\alpha=2\cdot\beta$ is not at all the same as saying $\alpha=\beta+\beta$, since $\beta$ copies of $2$ is not the same order type as $2$ copies of $\beta$, a phenomenon at the heart of the non-commutativity of ordinal multiplication. )

To explain the idea to a child, I would focus on the principal idea: whether finite or infinite, a number is even when it can be divided into pairs. For finite sets, this is the same as the ability to divide the set into two sets of equal size, since one may consider the first element of each pair and the second element of each pair. In the infinite context, as others have noted, there are numerous concepts of infinity, each with its own concept of even and odd. In my experience with children, one of the easiest-to-grasp concepts of infinity is provided by the transfinite ordinals, since it can be viewed as a continuation of the usual counting manner of children, but proceeding into the transfinite:

$$1,2,3,\cdots,\omega,\omega+1,\omega+2,\cdots,\omega+\omega=\omega\cdot2,\omega\cdot 2+1,\cdots,\omega\cdot 3,\cdots,\omega^2,\omega^2+1,\cdots,\omega^2+\omega,\cdots\cdots$$

This concept of infinity is attractive to children, because they can learn to count into the infinite this way. Also, this concept of infinity has one of the most successful parity concepts, since one maintains the even/odd pattern into the transfinite. The smallest infinity $\omega$ is even, $\omega+1$ is odd, $\omega+2$ is even and so on. Every limit ordinal is even, and then it repeats even/odd up to the next limit ordinal.

See the Wikipedia entries on transfinite ordinals and ordinal arithmetic for more information about the ordinals.