PrivatAir is still very close aligned with Lufthansa in an operational sense. LH is still manning a lot of the PA trans-atlantic flights with their crews since the codeshare agreement is still valid. But this only applies to the skedflights on behalf of LH. So if you don't wear the yellow LH ID-tag you don't even have a slim chance to make it onto their flightdecks.
It may get better once LH feels the need to take all crews to man their A380s which will require a huge concentration of manpower who are then needed to fly the PA flights.
Nevertheless, you may feel free to mail an application to PA. For the times after the LH domination. They will keep it on file if you come along with appropriate qualification.