TX does recognize "informal" marriage. Under § 2.401 of the Texas Family Code, an informal marriage can be established either by declaration (registering at the county courthouse without having a ceremony), or by meeting a three-prong test showing evidence of (1) an agreement to be married; (2) cohabitation in Texas; and (3) representation to others that the parties are married. A 1995 update adds an evidentiary presumption that there was no marriage if no suit for proof of marriage is filed within two years of the date the parties separated and ceased living together. I hope this helps, good luck.
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Short answer? Yes.
However, common law marriage is not as simple as just claiming it, they would have to meet the elements and you can attempt to disprove these same elements to show that they are not common law married.
They have to agree to be married, then live together as husband and wife, and hold themselves out to the public as husband and wife. If you can get him or her to testify that they have not done one of these elements, then you essentially disprove the common law marriage.
The father of your child is not married at "common law" by simply so claiming. However, even if all requirements of a "common law" marriage were met it would not give his "common law" wife any parental rights with respect to your child. The term step parent by definition means that they have no legal relationship with the child.