Yes. Especially true at other schools that own their health systems. Yes too at Pitt from an overall viewpoint, but there is a technicality since UPMC is legally separate. Patient care aspect is handled by UPMC, which owns University of Pittsburgh Physicians (UPP), but the academic appointments (conducting the research, teaching, or other university service duties) are under the university itself through the faculty appointments within the school of medicine or other health science schools. So these individuals, at the Pitt med campus and other Pitt research facilities in the area, are often employees of both UPMC (UPP clinical practice) and Pitt (academic/research), although UPP and Pitt share a common paymaster. In fact, some faculty are also employed by the VA for clinical practice there. But correct, for some of them, most of their time is in clinical practice. For others, more of their time is in research. And most pay acomes from either revenue from clinical practice revenue or research grants.
As an aside, because I've seen this come up before, note that ALL externally-funded research money flows to the university, not UPMC. This arrangement with the Pitt physician practice plan being owned by UPMC but the faculty appointment being done at Pitt, and all research being managed at Pitt, is also why UPMC subsidizes the university to such a degree that it does, because that clinical income helps subsidize the academic support (e.g. faculty and staff and facility overhead). Other complexities are ownership of facilities: Pitt rents research space from some of UPMC's facilities, which UPMC rents clinical space from facilities that Pitt owns. The two entities share a health library, and also share a legally separate fundraising foundation. They are very intertwined. Another reason, when talking about tuition dollars, it makes little sense to talk about overall university numbers like totals of faculty and staff. It is way too complex to do so.