"Enjoy" is the obvious choice, but "explore" works even better in my opinion. It invites us to think about what it means to live a good life. "Explore" suggests that they hope their child will not have a limited experience, and will encounter the different facets of life. At least, this is my interpretation.

to explore can mean "to investigate fully or thoroughly". Here perhaps, it is being used to mean to "experience it fully" or to "make the most of it".

"Your new life is full of promise..." = "there are many possibilities", and they wish, that this person is able to find out about (explore) many/all of them. It might also means, that they wish the person to be active and not just "enjoy" it but to work, so many possibilities come true. But that is a guess as I do not know the context.

Maybe the parents simply believe that exploring (being curious, learning, growing) is more important than enjoying. Or maybe they think that in order to enjoy fully you first need to explore (so you can learn to enjoy)?

Hi! I think that "explore" is used because he/she has a "new life", and so there are many unknown things that the person is about to discover. For example, the person may have moved to another city/country or has a new job or will start a family soon. "Enjoy" might be used, too, but from my personal view of point that would imply that there will be only good things without obstacles; being happy and healthy will make the person more prepared for the not-so-happy things that could appear on the new life :)