I grew up under similar circumstances. No treats allowed in the house, bad tasting food, when treats were allowed they were severely restricted, when not restricted they were eaten with abandon. I grew up fatter than my sister. My sister could have been a model. I read my mom's diary from her pregnancy and birth of me and the first time I was breastfed my mother thought (and wrote in her diary) "what a little piggy". I think that this perhaps had more to do with it than what kinds of foods were available. A subtle expectation/pressure/loathing of me and my appetite. I never got as fat as a lot of kids these days, though.
Female, 5'3", 50, Max squat: 202.5lbs. Max deadlift: 225 x 3.