Can I get reimbursed for the Ubers?

People who employ you are not responsible for your transportation costs. Have you asked them where you are allowed to park? I find it hard to believe that there is no legal place for you to park in their area. Where I live, people need a permit to park on the street. However, there are guest passes (each good for 24 hours) which residents give out to their guests.

You should really ask your client where to park and they should be able to tell you. I ask everyone too. How about their driveway or garage?

I know it doesn't help now but for your next bookings I would recommend that you confirm there is parking available; if not, let the client know that you will need to increase the cost to cover transportation. For this booking it's likely too late but I would recommend you mention this to the client so that they can give the sitter this information up front so they don't incur unexpected expenses. You should also talk to an accountant to find out if you can claim the Uber rides as a deductible to help offset the cost. I am curious, though, if this is a sitting job (as opposed to drop in) why would you be going back and forth to your place? When I sit at someone's house I assume I won't be home; for all intents and purposes I live at the house where I'm sitting. I'm not sure what arrangements were made with client but most expect you to be there when sitting unless you've made pre-arrangements. Good luck!!! I hope it works out!