Not sure how you came up with the link between PRINTZ and PARKAN...
If go to Beider's surnames encyclopedia  you find PRINTZ is coming from female name SHPRINTS (sorry to disappoint you),  and also from prince.
SHPRITS was common in some towns in Bessarabia.

I understand that Jews changed their names, as my great great grandfather did,  and I do not know what that surname was.

Yefim