I don't think there is a simple criterion, no, assuming your parallelograms are arbitrary.
An approach less general than that suggested by HenrikRüping, but likely easier to code is this.
Check if one of the four vertices of $A$ is inside $B$.  If so, return _Yes_.
Next check each edge of $A$ for intersection with each edge of $B$.  If an edge intersection is
detected, return _Yes_.  If all these tests fail, return _No_.  How to perform the primitive
intersection tests is all over the web, including [here][1].


  [1]: http://cs.smith.edu/~jorourke/books/compgeom.html