It seems like there are multiple issues here. First, you say "a friend took my rescue dog," which would imply that you own the dog. If this is the case, you are entitled to "take this dog back" since it is your property. There are legal avenues you can pursue in order to accomplish this. You should contact an attorney to discuss these avenues.
Second, if the dog is co-owned by you and your friend, the situation is a bit more complicated. The answer to your initial question about enforcement of verbal agreements depends on a number of factors. The short answer is "sometimes." Verbal agreements can be enforceable depending on what the verbal agreement is for, the terms of the agreement, etc. Some agreements are required to be in writing to be enforceable. I'm not entirely sure what a "co-parent" arrangement for a dog entails. And with the facts you presented, it's difficult to determine what legal remedies are available to you.
If you are serious about reclaiming ownership of the dog, you should contact an attorney to discuss your options to accomplish this. It may be as simple as having an attorney send a letter to your friend, or you may need to pursue litigation (in which case you have to determine whether this will be worth your time, effort, and money to pursue).