Can they stay in the US without leaving the country because my 8-year old sister has Down Syndrome and is dependent on them?

No, they cannot stay legally and cannot legalize their status in the US because of their unlawful entry unless they qualify for 245i. As for the sister, if she is a US citizen, she would be able to stay in the US but because of her Down's syndrome and dependence on the parents, of course she would need to go with them, if they choose to leave the US.

Based on the facts you provided, there is one solution: Your brother can file an I-130 Petition, together with your parents' applications for adjustment of status. It is extremely important to do the I-130 Petition right, because, if it is denied, the case will become hard to save. If the I-130 Petition is approved, the applications for adjustment of status will be denied, and your parents will be summoned before the Immigration Court. There, they can ask the Immigration Judge for the relief called "cancellation of removal" It means that the Immigration Judge can give them green cards to avoid the exceptional hardship that your 8 years old sister would suffer if your parents are deported from the U.S. There is a serious risk in this process; and anyone who would tell you he/she can guarantee a positive result would be lying. The procedure is very difficult and highly technical; not many attorneys can handle it successfully; and trying to do it without an attorney is pretty much like performing a surgery after watching some do-it-yourself videos on YouTube. At this time and on the facts you provided, there is no other solution. It is possible that there are other options available to your parents. It is also possible that, for some reasons, one or both of your parents cannot apply for cancellation of removal in the Immigration Court. To consider these possibilities, an immigration attorney would have to ask your parents, literally, dozens of questions; it cannot be done on this forum. So you can tell your parents that they need a consultation with an immigration attorney.

Your question requires an attorney consultation. It is not a simple question that can be answered on this type of forum. There are many factors that would need to be considered and evaluated. I strongly suggest that you contact an experienced immigration attorney for a face-to-face consultation and give him/her all of the facts surrounding your situation. He/she would then be in a better position to analyze your case, advise you of your options and make sure that the necessary paperwork and documentation is presented so as not to delay the process.

Your parents cannot do anything unless the US government seeks to deport them. If that happens, they may be able to apply for cancellation of removal.