Unfortunately in English contract law, two wrongs do not make a right. Just because one party breaches a term of the contract, that does not give the other party the right to breach another term. Neither does it 'void' the contract.
So what can you do if someone breaches a contract? Well as pippin says, if you write something like this in the contract...
"In the event that payment for the services is not received within 14 days of the due date for that payment the childminder reserves the right to withdraw the services with 24 hours notice until the payment is made. The parents shall continue to be liabile for payment for the services during the period of withdrawal."
... you can do exactly that.
You can of course give them notice, but unless you have something similar to the above in the contract, or a termination clause written like...
"Either party may terminate the contract by giving notice of not less than one month, but in the event that any payment for the services is not made within 14 days of the due date for that payment the childminder may terminate the contract without notice."
... you will have to continue to provide the services while the debt mounts up during the notice period.
So is thebody's husband completely wrong, or just misquoted?
Well, if you decide to withdraw your services without having a term in the contract which allows you to, that is a breach. The parents may sue you for their losses caused by your breach, for example the cost of temporary child care. Now if you can show that, acting reasonably, you did not believe that you would ever be paid for the services, you have a defence against that claim.
So in general, you might be able to get away with it. But in this case it is going to be very difficult to show that it was reasonable to believe that: they have always paid you before, just late.
Secondly, if one party's breach of a contract is fundamental, it may be a 'repudiatory breach'. This is not the place for a full discussion of this concept, but
basically it does not apply here because again the parents have been paying you so you are not being deprived of essentially all of your benefit from the contract.