Sorry, it took me a while to sit down and write this. I will not tell you what to do in your situation, but I recently got through a major illness of my own. Perhaps my situation can help you figure out what to do. I posted some of it recently here if you want to read that too, but I'll try to expand on it a bit.
http://chins-n-hedgies.com/forums/showthread.php?t=46307
This is fairly long and personal, but it has to be to explain my situation, I think a couple extra paragraphs may give you and your parents more insight into my situation.
I have never had the best health, my childhood was filled with chronic infections, reoccurring step throat even after my tonsils were removed, mono bad enough I was bedridden for weeks, a hypothyroid problem (my thyroid is basically a bomb now), and chronic fatigue syndrome. Through it all I always had pets. My family always had at least two dogs and frequently other animals. When I went to college we though I was through with my issues. This was not the case. I began in a dorm and while I did have symptoms, we though my doctor and I were handleing it. At one point had to be rushed to the hospital at four am due to chest pains. I was a 19 non smoker, non-drinker and never did drugs. The best the ER could come up with after a bunch of tests was perhaps the tissue between my ribs was inflamed. I was put on vicodin due to the severity of pain, and was basically given free reign to use it as needed. (I was terrified because my Dad had warned my his family gets easily addicted, so I would be in pain in the fetal position before someone would shove one down my throat. I was scared of an addiction.) This began a series of visits to specialists. Each time we went somewhere, the doctor would do enough tests to rule out their area of medicine and suggest another. After going through many doctors we finally go to one of the best nerologist in the country. He finally said he had seen cases like mine before, but there was no official diagnosis. He said usually the symptom just faded away in a few months and never came back. During this time, I had been switched off vicodin to long term pain medicines (the only options are anti-anxiety or anti-depression ones right now) and I also had to drop some classes and I moved into an apartment on campus along with my boyfriend, and I stopped trying to work as well as do school and everything else.
After the end of the school year I had to try to keep my scholarship which required certain credit hours and grades. I had kept all A's but had needed to go down to 27 credit hours in the year instead of 30 because of issues the first term. At this point I thought I was doing better and that the symptoms were gone, like the neurologist suggested may happen. I managed to keep my scholarship, but my boyfriend who had been my primary support through everything was leaving for military basic and teach school training and would be gone for a year. This is when I got my chinchilla.
After extensively researching I found a pet I could have and I though would fit me best (dogs were not allowed in the apartment, even though I had one at my parents that had kinda been "mine" a little more than the rest of the family's before leaving for school) I searched and found a person on here rehoming their little one. Yuki came with all supplies and his owner didnt want money, just a good home. I adopted Yuki, knowing I wouldnt be good in an apartment by myself. He helped me through the ups and downs and stress of school. I ended up in the hospital a few more times, unluckily right in the middle of final exam week this time. I was forced to medically withdraw from school.
Fast forward a while. We still dont know whats wrong with my officially. I occasionally have issues to new meds, or I get severely depressed because of how my illness has effected my life (I cant join the military or complete schooling to be a vet, at least not right now). My boyfriend is back from leaning how to fix airplanes and we have moved to a small house. Recently my body got sick and forced me to sleep for about 22 hours a day for a few weeks before the doctors figured out how to adjust my medicines again. But overall Im figuring out life with a weird chronic illness and enjoying life with my chin- we are considering getting more as well. Yuki bonded with me a lot while my boyfriend was gone, so now we're looking for "his".
With all my issues, yes, Yuki's cage occasionally got cleaned a few days late. He occasionally didnt get playtime or had to chew his hay off a cube instead of get it from the bag of fresher hay. He is never seriously effected by everything though. I know he's taken care of and spoiled far better by me than he would be in most homes. I have the added bonus that I have written out extremely detailed care instructions and a couple completely trustworthy people that I've "trained" as well as my boyfriend who is involved in day-to-day care with me. My list is pages long and includes even tiny details like where I buy supplies-just in case something happens. If I couldnt do something, there was plenty of backup. I know that my home is the best one for Yuki.
I also know Yuki is one of the greatest blessings I've gotten though all this. I love dogs and honestly regret my chin cant cuddle with me in bed. However I know I haven't had the energy to go out an poop scoop or exercise a dog. Yuki has been perfect to motivate me to actually get up and move on the days Im exhausted. I know sometimes I need taking care of, but Im an animal person- something in me needs to take care of something else. I have his cage set up where under a minute of sweeping with a hand brush and pan clean up all but the bottom of his cage. And the fleece there gets pulled and washed as needed. I can spend about a minute a day to take care of him for everything except socialization (he does have a big cage and a ChinSpin). On the worst days he can have a radio or tv, if Im a bit better, I can sit on a chair next to the cage and read or watch tv while he cuddles me through the bars. On good days Yuki gets playtime in his bedroom until he quits or I notice he's overdoing it. I couldn't have gotten though my illness with out a pet. I know my soul would have given up all hope. Yuki gave me the strength to pull through, because Im the best person to care for him. Im pretty sure I would have withered away at some point if it weren't for "hey I have to get up and use my muscles to go see Yuki today"
That's my situation. For me, a chinchilla helped me particularly and my illness allowed me to continue giving Yuki very good care, while inspiring me to fight my illness. Both of us benefited. I dont know your particulars, so I dont know if the chin is suffering due to your illness, or you are suffering due to the chins presence.
I second that another chin isn't a great idea right now. There's more work in getting and introducing chins that you are probably up for right now. It can take months to do a intro right and doesn't always work.
I think it may be a good idea to show your parents this whole thread as well. It may show them how you are trying to do the best for both you and the chin and they can see how others have responded.