Intersection of two deterministic parity automata
The following paper:
http://www.faculty.idc.ac.il/udiboker/files/AutomataTypes.pdf
shows that the intersection (and union) of deterministic parity automata may involve an exponential blowup. 

Thus, you cannot do much better than converting to nondeterministic Buchi automata, and taking the intersection there (and determinizing back to parity).