Can Company Not Give Promotion Because of Family?

Dear Experts, I work for a government agency. Despite winning several awards a year, I frequently get picked over for promotions. I am told the reason is that I have 3 children with autism and they feel my "family responsibilities" might interfere with my ability to handle increased responsibility. Yet, they dump more work on me than anyone else. I am the only provider for my family and I am feeling a lot of pressure at home and getting burned out on a job that doesn't give back. Help! Our Twitter Advice Project (T.A.P.) is no longer an active campaign. To find an answer to the above question, please use the "Search" box in the right-hand column of this website.