Why do we hear so little about Anne Boleyn’s mother? What happened to her mother and father (Thomas Boleyn) after Anne was executed? Is it true that Thomas Boleyn tried to save his own life by betraying Anne (by agreeing to her execution)?

I'm not sure why we don't hear more of Anne's mother because she was, afterall, a Howard. Elizabeth Boleyn (nee Howard) was the daughter of Thomas Howard, the 2nd Duke of Norfolk, and his wife Elizabeth Tilney, and she was a direct descendent of Edward I. In fact, all of Henry's six wives are descendents of Edward I - funny but true!
Elizabeth was a lady in waiting to Elizabeth of York, Henry VIII's mother, and his first wife, Catherine of Aragon, and it was rumoured that she had had an affair with Henry VIII, although this is now thought to be a myth and a mix up between the names Elizabeth Boleyn and Elizabeth Blount.
Elizabeth Boleyn died just two years after the executions of Anne and George and instead of being buried alongside her husband at St Peter's Church, Hever, her resting place is in the Howard aisle of Lambeth Church, which is now the Museum of Garden History. When I spoke to Elizabeth Norton about Elizabeth's resting place, she mentioned that it was traditional for a wife to be buried alongside her husband in the family plot, and the Boleyn's children who did not survive childhood are buried at Hever. One wonders, therefore, whether her burial at Lambeth was to do with an estrangement between Elizabeth and Thomas after the deaths of Anne and George.

Thomas Boleyn survived the fall of the Boleyn faction and was able to regain royal favour after a time. He was present at the Christening of Edward VI, helped squash the Pilgrimage of Grace and when he died in March 1539 Henry VIII ordered masses to be said for his soul. I don't think that he betrayed Anne and George, and that he agreed to their executions, I think he just survived the whole affair by doing nothing. Many criticise Thomas Boleyn for manipulating his children and then turning his back on them but I think he was just a product of his time, a man looking for royal favour, basking in the glory of royal favour when he had it and then keeping his head down when things went tragically wrong. He was not an evil man, just a survivor.