It looks like the basic question is: given two blocks, what is the number of ways of putting one on top of the other with at least one overlap?
Now, for some (small) given set of blocks, one can figure this out, just by counting. This looks like what you've been doing so far. More mathematically satisfying would be to come up with some formula for this number, given some representation of two blocks. This is actually pretty hard, getting you into the area of combinatorics. This can be graduate school stuff, certainly not first grade.
I'm assuming that's why you've simplified the question, just to get a simple formula for predicting whether the count is odd or even. In other words, is it divisible by two? My guess would be that the formula for the actual count is sufficiently complicated that the number for any pair of blocks would essentially be "random" in terms of whether the count is divisible by two. So there isn't going to be an easy way here.
Potentially there's an interesting project here in terms of seeing how numbers of possibilities can increase very rapidly. Rather than looking at oddness or evenness, you could look at how the number of possibilities grows as the two blocks get larger. You'll see an exponential growth, for sure. This could be plotted on a graph.