Here are the THS T&Cs
5.3. The following applies only to Sitters. You will:
5.3.5. pay any veterinary expenses up front and request the Pet Parent to reimburse you for all costs incurred;
5.3.6. immediately contact the Pet Parent in the event of any damage to property and get their approval for repair. If, in the event of an emergency, you are unable to contact the Pet Parent, you will pay for the repair of any damage caused, and request the Pet Parent reimburse you for all costs incurred (provided you have not caused the damage yourself)
and here is a post where these T&Cs have been discussed previously
https://forum.trustedhousesitters.com/t/sitter-has-to-pay-veterinary-expenses-up-front
https://forum.trustedhousesitters.com/t/emergency-medical-bills-additional-thing-to-consider