What do the words in parentheses mean? Should I remove or replace those?
The words in the parentheses or quotation marks act as defining terms so that the reader knows that when they see the same word listed in the document it refers to the person or organization whose name is entered in the space immediately preceding the parenthetical.
This Fitness Agreement (“Agreement”) is entered into as of the January 18, 2019, by and between Burpee Gym, LLC, a Virginia Limited Liability Company with a primary address of 111 Burpee Street, Photographsville, PA, 22222 (“Fitness Professional”), and Jane Client with a primary contact address of 111 Client Street, Clientsville, PA 11111 (“Client”), each a “Party” and collectively the “Parties.”
Do not touch the part that says "Fitness Professional" or "Parties". Anything in quotations like this is a term that is being defined within the contract. If you notice, you just outlined the name and entity type in the blank before it. That means the term Photographer = Your Business Name, Entity type. See example above.