So got into a discussion with a friend and was wondering. 

Is it possible for teiflings to have a demon parent instead of a devil or it is it solo devil?

The answer is "ask your DM".  Especially since tieflings don't have devil parents in the first place; their parentage is (generally) entirely mortal, the fiendish taint in their blood is from well more than one generation back.  A half-devil is typically a cambion, not a tiefling.  It takes a lot of thinning out of devil's blood for the resulting critters to not be awful, according to standard D&D lore.

That said?  Who cares what standard D&D lore says.  If you like the idea of your tiffle being the child of a demon of the Abyss rather than descended from long-ago pacts/"pacts" with devils?  Go for it.  If the DM's down and you're not fishing for random powergaming crud, there's no reason to say no.  

There was an Unearthed Arcana featuring Abyssal Tieflings, but it's from all the way back in 2015 and hasn't been touched on since.

Hm.  I can see why that didn't test well.  Players almost never care for randomization, even if it makes sense from a place as fundamentally chaotic as the Abyss.  I'm not overtly familiar with the fine mechanical differences between demons and devils - the Law vs. Chaos thing yes, general trendencies in the Monster Manual no - but if I were going to create an Abyssal tiffle...hmm.

Switching the point of Int for a point of Constitution, yes.  Probably trade out the spellcasting altogether; demons are engines of unfettered destruction, they don't often have the discipline for spells.  An easy way to do it would be to trade Infernal Legacy for the Relentless Endurance and Savage Attacks traits from half-orcs.  Those traits can reflect a demon's unstoppable drive for destruction as much as an orc's...well, unstoppable drive for destruction.  A once-a-day Death ward and a minor boost to crits seems a fair trade for a few useful once-a-day spells and a cantrip.  And obviously, switch Infernal for Abyssal, provided the critter has any reason to know demon-tongue at all.  Always bothered me that tiffles knew how to speak devil-tongue despite most of them having no reason whatsoever to know how to talk to monsters from the Nine Hells.

Quote from Yurei1453 >>

The answer is "ask your DM".  Especially since tieflings don't have devil parents in the first place; their parentage is (generally) entirely mortal, the fiendish taint in their blood is from well more than one generation back.  A half-devil is typically a cambion, not a tiefling.  It takes a lot of thinning out of devil's blood for the resulting critters to not be awful, according to standard D&D lore.

That said?  Who cares what standard D&D lore says.  If you like the idea of your tiffle being the child of a demon of the Abyss rather than descended from long-ago pacts/"pacts" with devils?  Go for it.  If the DM's down and you're not fishing for random powergaming crud, there's no reason to say no.  

This. 

For reference, in Planescape (where tieflings were introduced) it was tied to having blood from somone in the lower planes.  So if you want to judge by what has happened before, then yes there are tieflings related to Tanar'ri, Yugoloths, and so on.  

Mechanically, you could use any of the base Tieflings, and only change the language from Infernal to Abyssal in the character origin settings and call it done.  I have seen this in multiple campaigns.

I really don't understand why WoTC decided to 'simplify' this from earlier editions.

I agree with those points made above. I think if you're just looking to "reflavour" your Tiefling, then most DMs probably would be fine with it. But every table is different. I think at the point that the PC is a Tiefling and not a Cambion, the bloodline is so watered down that the differences between a Devil or a Demon ancestor are negligible. To some prime material people they would be considered the same thing. To others, it could be interesting to take pride in your infernal heritage only to learn later on that it's actually abyssal and perhaps that doesn't go over so well. You could play with it for sure. 

Quote from Yurei1453 >>

Always bothered me that tiffles knew how to speak devil-tongue despite most of them having no reason whatsoever to know how to talk to monsters from the Nine Hells.

 D&D treats language as if it's somehow genetic.

Oh I know, and it's never really not irritated me.  One of the pieces of Tasha's Cauldron I tend to use most is fixing origin languages.  Hell, just this last week I was working up an Umbragen drow for an Eberron campaign I'm looking forward to digging into, and one of the things Wizards actually did right in the Eberron book was a table of 'Common and Exotic Languages of Eberron', sorted by where and by who they were most often spoken.  1.) every single god damn setting book should have that table, and 2.) itr made it very easy for me to determine that Misty spoke Giant (the "Common" of her home continent of Xen'drik) and Daelkyr (the language of her enemies), but not Elvish.

...and also Common because I'm not gonna be that girl and play the character that doesn't speak a single language the party does, but yeah.  I've honestly found that fitzing with starter languages is one of the better ways of selling a character that was born/raised somewhere other than the 'Normal' spot for their species.  Swapping out Infernal for Abyssal frankly might be all somebody needs to do to sell the table on a demonic, rather than devilish, tiffle.

I don't know how common it is, but folks do swap out Abyssal for Infernal and call it a Tiefling.  You probably couldn't use the Tiefling variants (though the variant Tiefling is probably ok) tied to specific Archdevils, but there are demonic boons you could water down and come up with comparable racial traits.  

I'll minimize my usual soap box on 5e never claiming Tieflings have actual Devil parentage, but the expression of Tiefling features in a family tree is evidence/stigma of infernal transactions, probably dealings involving souls or other devil's contracts.  Which is another factor into why Abyssal Tieflings may not be a thing.  

I could see Grazz't and Frauz'Ur'luu's being able to use magic, a lot of demons are destructive onslaughters but others practice more intimate forms of destruction and I could see have a spell casting capacity.  

The language thing is a bit calloused, especially given 5e's position in re: my soap box and Tiefling origin.  That said, a work around I sorta use involves the idea that like some of the infernal magic Tieflings can invoke, Infernal, the language of damnation is something in the Tiefling's blood.  It's like something akin to a voice parallel to their "common" stream of conscious as they grow into recognizing their origins.  In most default settings, Tieflings choose not to listen to it, but they recognize it when they see and hear it, and know how to talk back to it if the opportunity presents itself.  I kinda think of movies like the Omen and stuff in this regard, Hell is sorta always there for the Tiefling to talk to (especially if someone wants to eat up the Asmodeus claiming all as his) but most keep themselves excommunicated, unless a party needs some help reading a particular bad book or inscription.  

In previous editions, demons had more innate spellcasting ability than they do now.  But really, I'm fine using any tiefling variant and saying it's got ties to some demon or other rather than devils.  Demons are creatures of literal chaos, after all, so they can have all sorts of random results when it comes to whatever passes for genetics for them.

IIRC, in the lore of previous editions, tieflings were just people touched by any of the lower planes, not just devils/the hells. So yeah. Just make sure your DM isn't against the idea or the like.