Texas law on parking offenses is Ch. 682 of the Transportation code. After you work through the sections on creating ordinances and having a hearing, we get to enforcement in §682.010 – you have been ordered, at the hearing, to pay a fine. Enforcement can be by
(1) impounding the vehicle if the offender has committed three or
more vehicle parking or stopping offenses in a calendar year;
(2) placing a device on the vehicle that prohibits movement of the
motor vehicle;
(3) imposing an additional fine if the original fine is not paid
within a specified time;
(4) denying issuance of or revoking a parking or operating permit, as
applicable; or
(5) filing an action to collect the fine, cost, or fee in a court of
competent jurisdiction.
(b) An action to collect a fine, cost, or fee under Subsection (a)(5)
must be brought:
(1) in the name of the municipality served by the hearing officer;
and
(2) in a county in which all or part of that municipality is located.
The first 4 options do not require further legal process, they just do it (small claims court cannot order seizure of a vehicle). To get money from you, they have to go to court, and it can be any court in that county. The Houston ordinance does not add anything beyond saying what kind of parking is forbidden and what the fine is.
Texas law doesn't go beyond saying "court of competent jurisdiction". Small claims court functions to make a basic legal determination that a party owes money, but that determination has already been made in the case of a traffic fine. Enforcement would be obtained through justice court.
