I don't think parchment paper would work well in an inkjet printer. The ink would just smear, since the parchment paper is designed not to hold on to stuff. Same might apply to laser printers, but I'm not as sure about that.
Usually people use freezer paper, typing paper or vellum (tracing paper) through printers for patterns. People also send fabric through printers, for direct printing on the fabric (usually, I think, fused to freezer paper for stability).
I haven't done any of these things. Just been wanting to! So, I could be wrong.....