Can a tenant bring an animal into a home without the lease holders permission

OK, the lease is in YOUR name, it either permits pets or not. Even if the lease permits pets, if there is a written lease to the subtenant giving you the right to allow pets or not, it is still your choice. If the sublease is silent about pets, you are still the tenant of record. The landlord has no right to alter the lease without your written permission, nor does your subtenant. Best to put it in writing to both LL and subtenant that you don't want pets and, if for any reason pets do appear, you reserve a right to charge a cleaning fee on moveout and an additional monthly fee to the subtenant for damage and cleaning. Good luck.