Princess Anne was part of the of the British Olympic team at the 1976 Games, and her father Mark Phillips won team gold in Munich in 1972 and silver in Seoul in 1988.
But Zara admitted that she hoped she was "luckier" than her mum, when she competes in the London Olympics next month.
"My mum fell off and I don't think my dad ran the cross-country," Zara said, referring to the jumping and endurance test which is one of the three phases of the sport of eventing.
She added that her father had not said much about his own experiences. "He's not a big talker. He's more interested in training."
Zara looked excited as she was unveiled with the rest of Britain's equestrian squad at a news conference this week.
"I just count myself very lucky to be riding on a team with these guys," she said.
At 31, the Queen's granddaughter is the youngest on the side. Riding with her mount high Kingdom, this will be her first ever Olympic Games.
She will stay in the athletes' village during the Games and when asked how she felt about it she said: "Obviously it will be great to be part of the Olympics and to get the atmosphere and the buzz of being part of it."