just a tad correction- they are her foster parents - her real parents are Padme and Anakin ( Darth Vader) Skywalker. Her real name ( birth name ) is Leia Amidala Skywalker.
Her foster parents queen Breha and Bail Organa of Alderaan adopted her right after Padme died. As we find out in ROTJ she always knew that there was someone out there that she was connected to or that she had a brother according to her intuition.Her foster parents died suddenly - But she knew that her destiny and true path in life was still to come.Being part of the rebellion, she knew there could or would be serious consequences as did her foster father who was also a politician that fought the Empire.
I hope that made sense. She Im sure mourned them in private, but knew something else in the future was waiting for her and that she would be of great importance. Some things in the future would be happy like winning the battle against the Empire and finding her twin Brother, plus falling in love. Others would be scary like learning Darth Vader is her real father or being Jabba the Hutt's sex slave. Her life was a non stop adventure of ups and downs.
if you mean her adopted parents,I'm sure she did .If you are referring to her biological parents,she hardly knew them.As she said to Luke in ROTJ,she knew a little bit of her mom(Padme),just imagens and feelings.She never knew or saw her real father(Anakin)...unless you count that brief encounter from estrela Wars:A New Hope as Darth Vader.
depende on who you mean for "parents" if you mean her adoptive parens, she had no time to mourn over her parents, even though they died blown up por the death star, and if you mean darth vader as her "dad" she always hated him, since he was like a Himler to Darth Sidious who represents Hitler