Have you found the marriage of Thomas and Rachel? This might (no guarantee) provide details of Thomas' father - name and occupation.
Cannot see a Carmarthenshire marriage that "fits" but there are two possible Merthyr Tydfil registrations -
Thomas Williams, Sep Qtr 1859, Merthyr Tydfil, 11a 437 - on same page Rachel Thomas.
Thomas Williams, Mar Qtr 1861, Merthyr Tydfil, 11a 380 - on same page Rachel Thomas.