Can I drill into the vise or the parallels?

Please don’t! Damaging the vise and parallels could prevent them from working as intended. Be cognizant in setting up the work so you are always drilling in between or away from the vise. If you have a through hole or other feature that is very close to the edge of the vice, you can use thinner parallels or a different workholding setup – that way, you can support your part without machining into the parallels. Soft jaws can also be useful if you want a specific workholding setup.