My reputable dental insurance provided a list of many local professionals who identify as periodontists but are not american academy of periodontology members. should i pay more (out of pocket) for aap members? or can i trust my insurance company?

Trust?: Insurance companies have little or no interest in finding you the best dental specialists. They check on basic credentials generally. I would combine both your own independent search of highly competent and respected specialists in your area and hopefully at least one will accept your insurance.

Not so reputable: If Insurance company lists a generalist as a specialist, they are doing so to make their product look more attractive to their clients. Remember that an insurance company's prime responsibility is fiduciary, not health care. There are good and just reasons for seeking out a specialist to provide specialty care. Do your research.