If you are going to list or lease real property, or attempt to engage in the negotiation of an agreement pertaining to such in Texas, you need a real estate brokers license.
There are businesses and individuals in Texas that call themselves "property managers," but they are really just glorified service and maintenance providers that are contracted by an owner or agent to handle repairs and upkeep only -- and the owner and/or licensed agent handles all the interactions with the tenant and all the legal agreements and processes that require a real estate license. Of course, contracting to performing certain types of residential maintenance work may require other types of licenses (plumbing, electrical, etc.).
So it really depends on what you mean by "property management" -- the specific services that the business provides and how it provides them.
(I'm not an attorney, so don't interpret any of this as legal advice. I am a business owner with property in Texas trying to offer a no-cost answer.)