The title gives it away. Though I'm not sure that there is any concrete evidence that they have names, at least the show never gave them names.
So what are their names in the fandom? I've heard Twilight's mom to be Star Sparkle or Twilight Velvet. And her dad is something like Crescent. But who knows... anyone got a consensus on this?

They don't have canon names

The most common names are Night Light and Twilight Velvet

958287 the gameloft game says Twilight Velvet and Twilight's Dad.

958287 mom, Twilight Velvet; dad, Night Light. at lest in most the fics i see 'em in.

The game calls them Twilight Velvet and "Twilight's Dad". Of course, they also call her: "Fleur Dis Lee" so yeah...

MLP Wikia lists them as Twilight Velvet (Mom) and Crescent (Dad), and that's as close to canon as we have.

According to the wiki, Twilight's mother was released in the blind bags as Twilight Velvet, and her father shares the same color scheme as a german g1 pony named Nachtlicht(Night Light)

Ooof. I seems like Twilight's dad really got shafted in the name department by the fandom.
Hmmm... interesting that Twilight's mom shares part of her name with her daughter. Maybe Twilight's dad shares a portion of that name as well?
I guess Shining Armour would be the odd one out though. <-- favorite child

That's fairly common naming practice with horses. You have one named "My Valentine" and another named "Matter of Opinion" and the foal gets registered as "My Opinion".

958287 Twilight Velvet and Night light (Nite Lite?)

958310

The reason behind that is Twilight's mom is a copy-paste of G1 Twilight, so they had to keep the name.

There's a story somewhere that pokes fun at how "Twilight Velvet" has implications that "Twilight" is the family name, and 's "First" name is actually Sparkle.

In Traveling Tutor, I used Twilight Velvet and Night Light, with rough personalities of Twilight Velvet being an older version of her daughter, and Night Light being the cool, calm, reserved anchor type, much like Shining Armor. If you want to take a look, it's chapter 20 and 21.

958287 Generic background pony #36482 and #5372

I am pretty sure, but don't quote me on this. Their names are: Doogle McLonglog, and Betsey Slaphappy. I may be wrong though....doesn't seem very "Pony" now that I think about it.

Night Light is her dad and her mom is Twilight Velvet, according to fanon. But I prefer to call her mum just Velvet. The two Twilights confuse me.

Her dad is Mr. Sparkle, and there's really no argument to the contrary that I'll accept.

958287 Read the fanfic Young, Wild, and Free. It has the answer

I believe this was already discussed in a thread in this group.

That being said most of the time the fandom calls them Twilight Velvet and Night Light based off toy information and designs. Twilight's mom is similar to the G1 Twilight.

I did see at least two fics that called them Dazzle and Orion.

Twilight Velvet seems the big one for Twilight Sparkle's Mom. I always referred to the Dad as "Shining Crescent" to give a little Father to Son naming conventions too.