How did Richard Nixon’s parents react to him becoming President?

As others point out, they were dead by 1968. However Hannah Nixon lived until 1967, and Frank Nixon until 1956. So they saw him Vice-President, and she fretted over how he looked in the Kennedy–Nixon debates.

How they reacted to him making Vice-President would be evidence enough of how they would have reacted to him becoming President. They were proud, of course. The way Nixon recounts it in his Memoirs, both parents encouraged him on no matter what: Frank from his death bed encouraged him to fight back against Harold Stassen during the California primaries; and while Hannah convalesced from an operation, Richard told her not to give up, only to be told by her never to give up—she’d just read an article about him being in the political wilderness.

Richard Nixon was a maudlin kind of a guy, but I believe the anecdotes: I wouldn’t have thought either is implausible behaviour for parents proud of their son’s political success.

There’s a moment in Oliver Stone’s Nixon when Hannah is asked by a reporter how she feels, and she says something frosty about her main concern being that he does right by God. That’s intended as foreshadowing, but the film is meticulously researched, and I do remember it sounding familiar. I haven’t found a trace in the Memoirs or in Ambrose’s biography; but Nixon reproduces the note Hannah gave him in the family bible on his Vice-Presidential inauguration, which he kept in his wallet for the rest of his life. Without the frostiness, the note says the same thing as the movie Hannah:

“You have gone far and we are proud of you always—I know that you will keep your relationship with your Maker as it should be for after all that, as you must know, is the most important thing in life.”