Only if the bylaws are written *very* poorly. The current trend is to expressly ban everything, then conditionally permit a limited number of specified critters, and has been for some time. Maybe, if you are in a really old HOA, you might get lucky. Its unlikely.
Remember, when it comes to HOAs, as voluntary contractual associations, HOAs generally trump State, County, and Municipality ordinances, insofar as they may be **FAR** more restrictive than local zoning would otherwise allow. In essence, every member gives every other member the right to have...