Hello sweetie, welcome welcome! We are routing so hard for you, we just want you to get healthy and have some hope for the future so very badly. Are you on board for the long haul? Aren't you so worth it? If you say no then that's where you've got to start with this.
At any rate, you have so much on your plate dealing with your weight and the sad situation your parents have found themselves in. So very sorry to hear about that. It would be a huge load off of your mind if you knew with every pound lost you were one step closer to a healthy you.
As far as the exercise, I not being an expert but pretty well read and researched about weight loss, would think that you shouldn't worry too much about it right now if it hurts your legs. Do not waste your money on any cheap or even reasonably priced equipment because these structures are wobbly and cheap-feeling even on those in the low 100s, so not only are they not built well for thin people they will not be able to hold your weight. You must check the weight limits on these machines before buying, but with exercise equipment its always a case of you get what you pay for. I would suggest something really low impact just to get the blood flowing and the lungs more and more (building your cardiac conditioning/developing your heart muscle). Find some video that you can pop in after you've fed your parents that is low impact, such as beginners yoga maybe or something using the arms a lot, and marching, knee ups etc and other such motions rather than high impact (jumping etc). Just something to increase your heart rate without hurting you much or too grueling to turn you off to exercise.
Now what exactly is your eating plan? For starting, you can be very very flexible and do not need to count cals and all that complex stuff (as long as you start LEARNING about calorie content) as almost ANY cut in calories should get the weight moving off.
Best piece of advice is to keep coming here, even when you're f**king up royally and completely thrown in the towel. No one will judge you, but as you know, most Americans quit before we get our weight off, and we just don't want you to be in that statistic. Even maintenance takes a lot of effort so it is a life long struggle for most people to stay at a reasonable weight, but TOGETHER WE CAN DO THIS.