Well, relating to the show, each character has, at one point, made a remark about their parents.
Note: All quotes are relative. I can;t remember everything
Snotlout:
Dad - Spitelout: Pretty self explanatory, we know him from movies and tv show
Mom - unknown: Referenced in RttE s2e3 (i Think) when Snotlout talks about how he learned stitching from his mother.
Twins:
Dad - unknown: Reference in Riders of Berk episode 5(?). When the cast puts out the boots for boot night (seemingly something they do only for fathers, a note for later) Ruffnut holds a boot to Tuff and he says, "Ugh, why is his left always smellier than his right?"
Mom - unknown: Referenced in Dawn of the Dragon Racers short:
Ruff: "Well you know what they say, you can't make an omelette without a few broken legs."
Astrid: "Uh, that's not how it goes."
Ruff: "It isn't? We should give out some apologies."
Tuff: "Yeah starting with mom."
Fishlegs:
Dad - unknown: Another 'Boot Night' Reference. We assume he has a father because Fishlegs takes his boots out.
Mother - unknown: Referenced in Riders of Berk (episode name - 'Breakneck Bog') when Hiccup is looking for a chest containing something from Hiccup's mother, Hiccup takes Fishlegs and Astrid to a dangerous place (title of episode). Fishlegs being scared as always doesn't want to go, and bargains like so:
Fishlegs: "I could give you something from my mother, you know how you like her crab cakes, or I could GIVE you my mother."
Astrid:
Dad and Mom - unknown: The only reference I remember is from S2E1 of RttE (Team Astrid) when Astrid's house is hit and burned down, Gobber says:
"Your parents are fine lass, they were at the yak races..."
So we can assume they are both alive since He says parentS and not just "your dad" or "your mom"
Basically every Teen Dragon Rider has a parental reference in the show