Thomas was on hand to fire the ball home in the 52nd minute after linking up with Reece Styche and Rowan Liburd.
Hereford had chances to extend their lead, firstly when Liburd was denied from close range before Rhys Davies had a goal disallowed.
The Bulls took the lead in the 11th minute. Simeon Maye - who has signed for Hereford after impressing in trial - swung over a corner in the 11th minute, which Cullinane-Liburd swept home.
However, Bideford equalised 15 minutes later when Josh Wadham was on hand to convert a Lewis Walby pass
After equalising, Bideford had a chance to go ahead moments later, but Billy Tucker's free kick was pushed onto the bar by Hereford keeper Brandon Hall.
- Hereford: Hall, Thomas (Trialist 60), Riley, Jagger-Cane, Gowling, Cullinane-Liburd, Nicholson (Finn 71), Maye, Liburd, Styche (Bishop 71), Pope (Davies 60)
- Sub not used: Hill.