Does parking ticket warrants show on a criminal background check?

Yes it shows up on background check if you were arrested for not going to court

for your tickets. If they finger printed you it will show on record.

Added; I doubt that they will be criminal warrants and therefore probably, in all likliehood, will not show up. However - - you should be aware that the outstanding tickets MAY affect your drivers license. The DMV MAY suspend your license due to unpaid traffic violations.