My client wants me to take the dog to a park to walk them., is that ok?

If you don't mind driving the dog to a specific location, then, by all means do it. The clock starts when you arrive at the client's home and ends when you return the dog. Therefore, this would mean the dog doesn't get a full half-hour walk. Also you may want to consider a charge for use of your car.

Don't do it. I have my own insurance policy with more coverage than rovers (rovers liability for a walker/sitter is very low) and I won't take a dog to one. Way too risky.

You may want to consider charging this client a higher rate to cover your extra time and mileage