The answer depends on the message. Yes, it can be 100% appropriate to message owners about pets you've cared for, even using Rover messaging, by going to past clients (You do not need to text Rover clients unless they previously communicated that they prefer that).
I'd suggest write down your message on paper, set it aside, look at it again later and ask yourself (or any dog loving people you now) if they'd be completely okay with receiving that message or if any of it needs to be re-worded. If it's friendly, positive, upbeat, and personalized, referencing how you were thinking of their pet (mentioning pet's name) and hope all is well, chances are that would be well received.