My landlord said I could have pets and after I moved in the HOA said I cant. Can I get out of my lease?

Depends on what the lease states, if you signed the lease and then said by the way I have pets and the landlord said okay, then no you can not break the lease. If it is in the lease and the HOA had to approve the lease then yes you may be able to break the lease if the HOA does not back down, and moves to evict you.