Yes, they can. The CC&Rs & bylaws of the HOA are no joke, it's very important to read them before you purchase a condominium. Not all HOA's have that restriction, but many of them do. Further, even after the first year, there may be additional restrictions on renting out your condominium. For example they may only permit a certain amount of condominiums to be rented and if that percentage has been met, then you would have to put yours on a waiting list. The purpose of that may be to either maintain their approval for FHA & conforming financing, or just to maintain "pride of ownership" as studies have shown that owner occupants take better care of their properties than tenants do.
Shane Milne | Lending in all 50 states | NMLS #81195