I claim that if is even, Peter wins and if it is odd then Bob wins.

Case 1: is even. Then Peter makes a move to an adjacent square. Afterward, he tiles the remaining with dominoes. If Bob moves into one of the squares of a domino, Peter moves into the other one. In this way Peter's strategy is determined and he wins. After more turns, Peter fills in the last square.

Case 2: is odd. Then before Peter's move, Bob just tiles the white squares (excluding the top left) with dominos. This is obviously doable because a by and an by are both possible to do with dominos. Anyway, then if Peter moves in some tile, Bob just finishes the tile and then we are happy. Bob finishes the board and that's it.

So done in either case.