Say if I played an E major scale over a C major scale, would this type of harmony be what is known as parallel thirds?
And I know it wouldn't sound very nice, but one has to know the rules before they break them.
Branny is right, but parallel-ness only depends on the interval number, not quality. Just harmonizing a third up in key is still parallel 3rds.

And yeah, it would sound bad if you did that...it is probably a better example of polytonality than traditional harmony.