Was it mandatory to join the NSDAP during WWII?

Not really.

The NSDAP were fairly scrupulous in maintaining party membership as a volunteer process in which applications had to be filled out and signed by hand. This was unlike other organizations within the dictatorship like the Hitler Jugend or the Deutsche Arbeitsfront in which there was a degree of mandatory membership (although, it should be noted in in these compulsory cases there was a line between membership and active participation).

There was a considerable degree of voluntary joining into the ranks of the NSDAP in the early months of 1933. The "old fighters" of the NSDAP, i.e. those who joined the party prior to the party's political ascent in the early 1930s, termed these late-comers Märzgefallenen (March fallen) or Märzveilchen (March violets) as a form of scorn. This aspersion was a pun not only referencing a sort of sudden spring of NSDAP support in the March elections of 1933 as well as referencing the fallen 1848ers who died in street clashes in March 1848. The German left had sometimes used the term Märzgefallenen as a reference to fallen martyrs. The NSDAP's leadership was alarmed by what they feared was an opportunistic influx of new members and imposed a moratorium on new members in April/May 1933. There were exceptions to this membership ban. Germans involved in NSDAP youth organizations prior to 1933 were allowed to join the party as adults and the NSDAP actively courted business and cultural elites for whom the ban was no barrier to joining the party. The NSDAP lifted the ban in 1937, but it was still leery of opportunistic joiners and still retained a loose control over membership. It was only in 1939 that the party leadership felt comfortable enough to allow for more or less open enrollment into its ranks.

But party membership was still a voluntary affair. There were moves as the war deepened to encourage young Germans to join the NSDAP. This was done in the hopes that such ideological commitment would translate into military fervor once these youth were conscripted into the Wehrmacht. Historians of the Third Reich have noted that it is possible that some individuals had their party applications submitted without their knowledge. Local NSDAP leaders were under intense pressure to show results, especially among HJ units. But there is little evidence of unsigned applications being accepted.

Despite the lack of evidence some Germans have claimed that they have no memory of applying for party membership even though their party cards are in the archives. Some of this denial is no doubt an exculpatory narrative to cover up an embarrassing biographical fact (i.e. a lie). But it should also be stressed that youth indoctrinated within a dictatorship and facing the pressure of total war might have signed up without clear ideas of what they were getting into or what party membership entailed. Whatever the case, this narrative of being dragooned into the party and its organizations against one's will was a powerful one. Generational memory within Germany often saw children and grandchildren projecting narratives of victimhood and resistance on their parents and grandparents.

So the idea of being forced to join the NSDAP in 1942 is not really credible. But Kast's alibi for his father is well in keeping with postwar trends of denying parental engagement with the Nazis.