You mention they have base board heat and I'm assuming it's electric, which is by far the most inefficient way to heat. You say this is the first winter they have lived there, and if they moved out of a house with gas heat they certainly would get a rude awakening when they got their first electric bill, and it just so happens it was the same time you parked your rig there.
Maybe help them with the electric bill to keep them pacified, then when you move out and the electric bill doesn't go down (if they are still using heat) they will know it wasn't you.
Hopefully you don't move out in April when they stop using heat and they will think they are right and will have to wait until next winter to see the spike again.