You are not in a pickle. Your parents are, and they are trying to rope you in to rescue them from their poor decision-making processes.*
You should not do this. They are bad with money, let this get so terrible that they owe more than a third of what the property could even be sold for optimistically, and have terrible credit. They are extremely poor risk and I doubt they'd be able to pay you back at all, and they will likely end up losing the house eventually anyway, taking your contribution/equity with it.
If you give them money, they are bailed out, will not change and it will need additional triage in another year or so.
There are siblings, and an aunt and uncle living there with your parents? That is insane that this many people can't manage to collect enough to pay the mortgage, so the idea that you are their only hope is ludicrous. You are not. It is not the only path. You can let them be adults and figure out how to go forward without you providing your own hard-earned money to bail them out of a mess they made that could have avoided.
If you really want to help them, you should either gift them the amount so it's not a loan or equity since you almost certainly won't see it again and may not want this hanging over you (which no, sorry, I couldn't do this with their track record - this isn't an unexpected emergency; they would have had plenty of time to right this ship before it hit the iceberg), or find them a path to getting the house foreclosure stopped long enough to get it sold and pay off the mortgage, start budgeting, paying down their debt and learn how to live within their means in a small apartment or house that they can actually afford.
*The only reason I'd be slightly more inclined to give/lend a substantial sum of money would be if someone was VERY sick, had an accident or some other serious incident that caused them to have no choice but to neglect the mortgage so they could get the emergency under control. BUT most mortgage companies will work with people in dire circumstances and I'd like to think you'd have mentioned if they'd had something like this happen so we had more sympathy for their situation? But without further clarification, it seems as if they were just ignoring the mortgage for whatever reason (too much frivolous spending? too little earnings and inability/lack of interest to find work for all able-bodied adults? mental block on speaking with the mortgage company the instant they realized they were in trouble? stubboness about the dire circumstances that should have triggered them putting the house up for sale ASAP to get out of a situation they could not afford?) then got behind, and then let it get REALLY behind and didn't bother doing anything until they entered foreclosure - which is a huge mess and again, they'd have had many chances to fix this before that happened.