He has a magic sword that can, in one stroke, chop off either one head, two heads, one tail, or two tails.

This dragon is of a type related to the hydra; if one head is chopped off, a new head grows. In place of one tail, two new tails grow; in place of two tails, one new head grows; if two heads are chopped off, nothing grows.

What is the smallest number of strokes required to chop off all the dragon's heads and tails, thus killing it?

Yes, I wasn't sure if dead dragons could regrow parts or, as you said, a dragon would live for a short period so that its parts could regrow.  However, for my answer, I took the question to mean that, if all heads and tails were cut off, even if parts are pending to be grown, that the dragon is dead.