I work with first graders to learn the difference between odd and even numbers. Here is a typical way we will demonstrate that odd numbers have an "odd man out" and the even numbers have "all partners."
But we can also examine when odd and even numbers come up when we are counting with other patterns. I haven't done this before, but I think combining the concept of counting patterns with odd and even patterns could build understanding even better!
It is interesting to see that counting by 5s yields another odd/even/odd/even pattern while counting by 2s yields all even numbers. (This is an association I have previously shared as a fact, rather than allow students to explore and discover this for themselves.)