It's really hard to find somewhat accurate information on this guy. He seems to have been part of the White Army fighting the bolsheviks until the left for Yugoslavia. He taught there until 1941, when Germany invaded Yugoslavia. He stayed in the country until 1942 or 1945 (unclear), when he left for Bavaria. If the latter, it's possible he tried to avoid Operation Keelhaul which would have seen him forcibly repatriated to the Soviet Union.
After the war he was in an American-run camp for displaced German peoples, where he met Hans mother. They left for the US in 1950/1951 and settled in Huntsville. Since he was connected to the University of Alabama (and was a professor in sociology before the war), it seems likely that he was somewhat connected to Von Braun, who was also important in elevating the UoA.
Interestingly, it's said that Von Braun and his fellow German scientists were more progressive than the Americans there. Von Braun was threatened by the KKK for a while because he hired black people.
But the timeline is shaky. Hans claims that his dad fought communists in Russia and Yugoslavia (specifically Tito's), but it's unclear when he did so. If he left for Germany in 1942 he couldn't have fought Tito's communists because he wasn't around.
Presumably if he was around in this period, he likely due to his nobleman origin would have been part of the Chetniks, who had a complicated relationship with the Axis. Not quite Nazis, but plenty of war crimes to go around still.
But the most credible source I could find claims he left for Germany in 1942, seemingly making it impossible for him to have been part of the Chetniks. Which raises the question: who did he even fight, if anyone at all? He seems to have consistently moved towards areas where there was less or no fighting at the time (and that he could actually move to). It's not impossible that he tried to avoid the wars once they reached his doorstep. He was also an anti-communist, which at least could explain the move to Yugoslavia and to Germany if it happened in 1945.
I'm not sure if based on these fragile timelines we can conclusively say or suggest he was part of the Nazis, Chetniks or any other group. The gap in his resume could well be explained by the fact that it's an academic resume, likely leaving things off if they weren't relevant to his academic career. Had he worked in a factory in that time for example, it probably wouldn't have been listed either.
We do know that his son's a fascist shitstain though. Whether he himself was a Nazi is highly unclear given the contradicting timelines and statements.