I think it's unfair of a parent to ask to change their arrangment as a passing conversation at the front door in the am. So, try this next time a parent tries to catch you off guard.

"ok, let me look at my business plan and I will get back to you. Would you do me a favour and email me that request so I have it on file and can respond appropriately"

it buys you a little bit of time to decide. Changing your scheduling isn't something you should be able to do on the fly. I have spread sheets... business plans... budgets. All of these things I have to consult if someone wants to so much as take vacation in 3 weeks.

If you need these clients for the $$$ then you should agree for now, give them a $40 a day price point and start looking to replace. Once you find the right family give this one the term letter based on financials. You give them the choice at that point to either pay X amount for X amount of days regardless or find someone else.