Dogs definitely are not allowed at any pools, and I think that is very wise! But I do love the thought of your mom going to a Rec Centre and requesting a Guest Pass for your dog.
JillyBeans, you might be interested in taking your doggie to the private dog park, Doggie Doo Run Run, on CR 101. There are lots of really nice dog owners there if you're interested in socializing. There is a small dog area for dogs under 30 lbs. At first a timid dog is very anxious there, but they grow to love the place, and it helps them develop self confidence. The first time I visited DDRR, my little dog, Biggins, just sat in my lap. The second time he was flying around and playing, and now he can't wait for his daily visit there.