That question has probably hung over the entirety of the series. Let's look at the situation for each mane member of the six.
Twilight Sparkle: Her parents have made most likely as equal appearance/maybe less than Pinkie's but we are aware of her parents. Making an appearance in Cutie Mark Chronicles, The Canterlot Wedding - Part 2, Magical Mystery Cure, a slight appearance in Equestria Games, and most recently - and obviously, the second part of The Crystaling. Her fathers name is Night Light and her mothers name is Twilight Velvet.
Pinkie Pie: Her parents are, like I said mostly around as much as Twilight's, proving we are aware of her parents as well. Making their appearances in Cutie Mark Chronicles, Magic Duel, and Hearth Makers. Pinkie however mentions bringing her mother to the Grand Galloping Gala in Make New Friends but Keep Discord. Applejack also finds party plans for their anniversaries in Party Pooped too. Her fathers name is Igneous Rock Pie and her mothers name is Cloudy Quartz.
Rarity: Her parents are probably the third most known of in the fandom/show. They make appearances in Sisterhooves Social, Inspiration Manifestation, and her dad appears alone in One Bad Apple. Her fathers name is Hondo Flanks (Magnum) and her mothers name is Cookie Crumble (Betty Bouffant), however their names were never really mentioned anywhere in the show but in other places.
Rainbow Dash: Her parent(s) have rarely been touched upon in the show since only one parent has been seen. Her father has made only one appearance in Games Pony Play. His name is Rainbow Blaze which was not mentioned on the show but made official in Hasbro's MLP card game on his card.
Applejack: Her parents are not known but have been mentioned recently in Crusaders of the Lost Mark when she wishes her parents were there to see AppleBloom get her cutie mark. The statement either hinting that they passed away OR some other situation.
Fluttershy: Her parents have not been mentioned in the show, maybe in some works, but I have no information on family of Fluttershy, HOPEFULLY that changes!
Hope this was informal to somepony