The Park Police does have jurisdiction over the Capitol grounds and they were heavily involved in J6.
I am hesitant to give links because they seem to be left wing propaganda on J6.
Here is a neutral one on the duties of the Park Police:
https://www.nps.gov/subjects/uspp/jurisdiction-and-authority.htm
https://en.wikipedia.org/wiki/United_States_Park_Police
“In 1925, Congress placed the Park Police in the newly created Office of Public Buildings and Public Parks of the National Capital.”