What rights do children have in Missouri when the father dies without a will and the real estate is owned solely by him but he has remarried but property is still titled to father and mother?

So if I follow you: the deed was to your father and mother, mother is no longer married to father, new mother is not on the deed, and father died.

The divorce papers probably would have severed the first wife's title (in exchange for some other fair portion of your father's assets), as a matter of law, regardless of what the deed says. It is usually "cleaner" to file a quitclaim or release deed, but not always necessary.

Because there was no new deed naming the second wife, the wife may probably claim a statutory portion of the house as part of the estate, even if there were a will naming someone else. Because there is no will, the local laws of intestacy will apply and the surviving spouse usually has to split it with any surviving children (e.g., she gets half, the kids get the other half).