I'm beginning to think we're getting trolled.

"Honor thy father and thy mother."

Sure, but financial support gives material support to their sinful state of life (unless of course they are in dire straits, about to die, and no one else aids them).

Illegitimate children are not entitled to inheritance, but are they obliged to financially support their unmarried parents?

This is case specific and decisions are based on charity. Were the children raised up Catholic? Best example is John of Austria, illegitime son of Charles V. Charles recognized him and made sure he was educated a good Catholic. Even Philip II when he became king recognized John as half brother because John was honest and faithful, trustworthy to the point of  leading the battle of Lepanto.
Bottom line, this is not a black and white deal...

Sure, but financial support gives material support to their sinful state of life (unless of course they are in dire straits, about to die, and no one else aids them).

This is the only situation in which anyone is required to financially support their parents. As long as they have a roof over their heads and they aren't starving, you've fulfilled any obligation there.

"Honor thy father and thy mother."

Seems pretty clear-cut to me. If they are your parents, you are obliged to see they don't end up homeless or starving. It doesn't matter if they're saints, or in an irregular living situation (concubinage).
I understand what the OP was getting at: "I'm either your son or not. If I'm your son, I want a share of the inheritance. If I'm not, then I shouldn't have to take care of you in old age."
Seems kind of selfish to me...

I am not the OP however I have experience with this.  My parents were not in a valid marriage (civil remarriage without annulment).  My mother's divorce from her first husband was against her will (if that matters at all) and my parents didn't know each other when she was still with her first husband.  Anyway, me and my siblings are the product of the invalid situation.  Mom is fiercely independent, had a stellar career, two college degrees, and resented being burdened with children.  My brother was given away as a teen and she neglected me and my sister to follow her own pursuits.  She eventually divorced Dad because of his alcoholism. We kids are now all in our 50s.  Dad is deceased.  Mom lives (by her own choice) alone in sub-standard conditions in the small town I grew up in.   She squandered her inheritance from her parents in less than a decade (yes, it was hers to do with but she didn't even save or invest for her own future).  Now she's broke and likes to blames others for her situation (OK, boomer).  To put a cherry on top of it, she's a new ager and doesn't see a need for religion as she "goes directly to God" (the god inside of herself).  She never had us baptized despite her assertions that she is a Christian.  Need I say that she has no relationship with any of her kids?  And none of the three of us speak to each other because she saw to it that we all hated one another through her slanders and manipulations over the years. I practice No Contact and it's been a blessing.

Last year my financial situation improved and in the interest of doing penance for my own sins I decided to begin sending her money each month.  Only to honor the commandment. I have no interest in a relationship with her.  She has done great evils that I can't print here.  As expected I got a bizarre rant from her about QAnon, pizzagate, Jeff Sessions, the greatness of my nephews, her tears for the children of the world, but no mention of how my children might be doing. Typical.  I didn't respond but kept sending the money.  She now says she's on a waiting list for a senior living facility but it's a 5 year wait.  My brother has invited her to live with him but she declined.  She tried living with my sister but left after 2 weeks, stating my sister was forcing her to eat healthy food and she didn't like that control.  

I made a decision for God's law instead of my own resentments.  Even though I intellectually knew my actions were good it wasn't until recently that I felt good about it.  I have spoken to half a dozen priests about this and they unanimously agree....my obligation at this point is strictly financial support and to pray for her.  

Hope this helps.

Last year my financial situation improved and in the interest of doing penance for my own sins I decided to begin sending her money each month.

Alms that could've been spent on building a church, supporting a convent or monastery.