Can a protein kinase phosphorylate both a threonine and tyrosine residue?
Ony a few protein kinases can phosphorylate all three amino acids — these are classified as dual-specific kinases (EC 2.7.12.1). Examples are APK1 from Arabidopsis or MEK kinases in mammals.

As with other enzymes, the residues at the substrate binding site determine which substrates can be accomodated there. Structures or Ser- Thr- and Tyr-kinases are available for inspection, should you wish to explore them.