I personally need to think you need to terminate today, before the 3 weeks is up.
This family is NOT a good match for your home daycare (or pretty much any other). By going beyond the 3 weeks you forgo your right to keep the deposit. It doesn't matter what the parent verbally agrees t...if you terminate this family after 3 weeks you WILL LOSE your deposit because I can almost guarantee that they will take it as a personal insult and they will seek to make this hard on you. You both signed to the 3 weeks if you go past that they will be able to seek legal action against you.
They are clearly not comfortable with letting their child cry it out. They will not change anytime soon. Good on them to seek sleep therapy. we can only hope that who ever they hire has the knowledge and common sense to get the parents to help train the child to sleep on their own (and not some wonky person that just further solidifies their belief in letting the child sleep poorly to forgo any tears).
But at the end of the day this is not going to be a quick and easy process. it will take months and months of everyone slowly changing. You will NOT see progress on your end anytime soon.