According to the manual, A (or X on PS2/3) defaults to "Select Nearest Target/Confirm". Perhaps the pet is the closest thing to you?
http://na.finalfantasyxiv.com/lodest...de/win/manual/

As others have stated, use your arrow buttons instead. L1+Circle defaults to enemy targeting mode (if you've enabled the filters), and you just use left/right arrows to cycle enemies, up/down for party members (or you and pet if solo).