I'll be the first to admit I don't know much about .htaccess files. However, a couple days ago, I needed to use one just like the one you're talking about. So I researched it, and got it working.
From the looks of yours, it's just backwards. I think it should be like this:
Options +FollowSymLinks
RewriteEngine on
RewriteRule EditPet.php?ID=([0-9]+)$ /EditPet/$1
If you're wondering what i did, first I took out the ^ from EditPet.php (not sure if that's correct, but there may be a / before the E so it's probably better to just leave it out). Then I swapped the ([0-9]+)$ and $1, which makes sense if you think about it. You want to match when there's a digit in ID=whatever, not match when there's a digit in /EditPet/whatever.
Let me know how that works out.