Yes, in many ways. One is breach of contract. Two is bad references. Three is fraud. Give your notice now and see if they want you to stay on until the end of the 4 weeks. If they find someone sooner I guarantee you will be asked to leave earlier.
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As you've seen from the 2 previous answers, people's opinions can differ, with 1 person thinking there's a contract and another thinking there's not.
From your description, the parents would have a basis to CLAIM that you breached a contract by leaving without notice. They might not succeed in that claim, but you would have to fight a lawsuit to prove them wrong. There is almost nothing that someone in your situation can do to GUARANTEE that the parents won't take legal action against you. So don't look for absolutes.
What you want to do is consider the range of negative consequences, and prioritize them. It seems to me that being kicked out of the house before you have any place to go is about the worst potential consequence. Getting sued in a month, or six, or perhaps never is a less significant risk right now.
To eliminate the first risk and to try to minimize the second, I would suggest the following: Wait until you have your ride lined up, and talk to the parents shortly before. In your conversation, be honest, and (if you can honestly do it given whatever has transpired) offer to stay for the 4 weeks, or perhaps some lesser amount of time, to allow them to find a replacement. If they tell you to leave immediately, well, you're covered, and your offer to stay will probably constitute the 4 weeks notice the "contract" (if it exists) required you to give. If they ask you to stay, you have 4 weeks of pay while looking for a new job.
If the atmosphere is such that you could not possibly stay on for another 4 weeks, or even an extra day (and it may very well be the case if there's verbal abuse), then tell them Friday/Saturday, leave, and deal with whatever fallout comes later. No one should stay in an abusive situation out of concern that there might be litigation later.