hello...well i learned the hard way that scales are really not the best indicator to how 'healthy' your weight is ....i was at around 87lbs (i am 5'2) like you and really just sick looking, i recently went over some older photos and it just looks nasty, i went into treatment for a month, because of heart problems
but, just because you might gain weight does not mean that you have to in an unhealthy way....i hated hearing from my parents that i should eat something like fries, cookies, ice cream, whole milk, steak...just because i was underweight...i actually think that in a way food issues are similair in lots of eating disorders...whethers its bulimia/binge eating/anorexia....its a sense of control...lack of control...and regaining control..often using food as a physical manifestation of emotion...
anyways point it, if eating fried stuff makes you feel awful, dont eat it, if your parents think you need more body weight let them see that you are eating healthy, and more importantly that you are eating...
right now i am trying to put on a little muscle and fat, but rather than trying to do it quickly, like eating fried food and junk, i just want to put it on in a healthy way...exercise and eating right...
anyways good luck, and dont let the numbers on a scale run your whole life...its just not worth it