OK, for ALL professional services performed in the hospital, you bill each service under the provider who performed that service. So the physician acting as the surgeon will bill the surgery under their own credentials and the PA who assisted will bill their services, with the AS modifier, under the PA's credentials. There is no 'incident to' billing allowed for services performed in a hospital, so you can never use the physician's credentials to bill for the PA's work that was done in a hospital. (The only exception is a split/shared visit where both physician and PA perform a portion of an E/M service for that patient in the hospital on the same date, but that is a different thing and is not 'incident to').
In a physician's private office the physician may bill, under his or her own credentials, the services performed by his or her own group's employees when those services meet the definition of 'incident to', which means primarily that those services are performed to carry out the physician's plan of care for the patient, and which are performed under the direct supervision of the physician (i.e. the physician is in the office and available to assist if needed when the services are performed). So when a PA, who is employed by the physician (or group) is fulfilling this function in the office and is not treating any new problems or making any independent decisions that alter the plan of care that the physician has ordered for the patient, then their services may be billed using the physician's credentials. But if any of those requirements aren't met, then the PA's work needs to be billed under their own credentials.
Does that help explain it better?