Rocky can sometimes be naughty. He jumps on my bed and tries to tear the sheets. I would
punish him by not playing with him. Even though Rocky is a big responsiblity, I still enjoy taking care of him.
Should I begin a new paragraph where it says
Even though Rocky is a big responsibilty, I still
enjoy taking care of him.
No, you are just continuing on with the same topic. Paragraphs are also generally longer than one or two short sentences apiece.
By the way, your 'would' should be deleted if you really do punish Rocky by not playing with him.