This is usually the first question asked at retirement community informational meetings. Yes of course they are allowed at active adult communities. Assisted living and continuing care communities may not allow pets but there are some that do.
Every community usually has a pet policy which states the types and sizes of pets allowed. This is usually consisting of normal pets dogs, cats, birds,and animals maintained in an aquarium and terrarium pets (not venomous). Size limits are usually small animals under 40 lbs.
Of course pet registration is required with their community and all pets must be up to date on their shots and immunizations. You must also comply with local county animal ordinances such as the leash law etc. The owner is also responsible for all actions of their pets(damages, disturbances etc).
If you would want to have more than two pets in a room you would have to get an approval from the board of directors.
Many senior care communities realize the positive effects of the companionship of animals with their owners and thus allow residents to have a companion.