Thomas Parker being a pretty common name makes finding things harder. But I'm pretty certain this is our guy.
Thomas Parker enlisted in the 2nd Rhode Island Volunteers June 5, 1861, aged 39. He was wounded 15 September 1862. I didn't find when this service ended. He reenlisted in the 2nd RI in December of 1863. September 1, 1864 he was promoted to corporal. That month on the 19th he was wounded again. He was discharged July 13, 1865. Then in January 1866 he enlisted in the 15th US Infantry, company G. That September it was reorganized and his company was transferred to the 24th US Infantry. He was discharged February 5, 1868.
He entered the US National Home for Disabled soldiers in Togus, Maine February 12, 1869. His place of residence is listed as Rutland, VT. They state his disability as "fracture right leg caused by a kick from a horse and general debility." His stay there lasted until May of 1870.
I haven't been able to turn him up in US Census records - dates for his wife's and daughter's births would help.