The parents of the Dalai Lama have traditionally held honored positions after his recognition.

The Fourth Dalai Lama's parents were Mongol Khaans. A position that was no doubt advantageous to the growing Gelukpa sect.

The family of the ("Great") Fifth Dalai Lama Lobsang Gyatso, was from the famous Zahor clan and was potentially chosen for their established renown and prestige.

The father of the Seventh Dalai Lama was clearly ambitious and was instrumental in plotting the downfall of the Tibetan King Polhanas (Pho-lha-neh) via the Manchu Emperor, and getting his son declared the temporal and spiritual ruler of Tibet. The Seventh Dalai Lama, Kelzang Gyatso, though technically the head of the Tibetan Government, spent most of his time dedicated to religious matters leaving others, like his father, to do the day-to-day running of the realm. Though that didn't last long and Kelzang's father was soon sent back to Beijing to be reprimanded. Luckily for him (the Manchu weren't known for their leniency...) he was sent back to Lithang, Kham (today in China's Sichuan province), after pledging an oath that he will no longer interfere in politics.

The other parents of a Dalai Lama that are known for their post-recognition activities are the current's (14th) parents. Tenzin Gyatso's mother was particularly renown as her son, the Fourteenth Dalai Lama, was only one of three tulkus (reincarnated lamas) to which she had given birth. His father, meanwhile, was soon propelled to a position of power in the Potala Palace while Tenzin took the vows of a novice monk and began his religious training.

The parents of recognized tulkus in general are usually the source of a lot of local religious devotion. A recognized child is often followed by locals presenting the parents and family with gifts of food and money, often for years.

Sources:

Kundun: A Biography of the Family of the Dalai Lama, Mary Craig

The Fourteen Dalai Lamas: A Sacred Legacy of Reincarnation, Glenn H. Mullin

Tibet: A History, Sam Van Schaik