I had to give up some ground with kids and gardens-- literally. How old are your kids? That will help you decide what to do.
Here's what I did: Give them the front foot or so of most every bed (we always had plenty of beds). Give as much as you can give. Let them dig and plant and play in it. Let them establish trails through the garden because it's fun when they are so little to get so lost in it.
Protect what's left. If they are young enough, they simply don't know any better than to pull up plants. Wow-- you can pull up this plant, but not that one! What's the difference?? This one has had time to establish its roots, and we need to give them time to do that, just like kids need time to grow up as well. You can see the difference between plants that have grown up and plants that haven't established. But at a very young age, this message will take a very long time to sink in. So, in the meantime, invest in inexpensive plants, and in ones too big to pull up. Save the rest for when they are older. (Sorry, but for me it's not worth the fight, and I still had a great garden because there was so much of it!)
Give them their own plants to plant and nurture. I remember my oldest when she was four would not stop picking the zinnias. It was good, abundant year, so what the hell. I wasn't going to argue, though I pointed out that if she waited a few days, the zinnias would get big and beautiful. Well, we went camping and came back a mere 3 days later, and in that short time the zinnias has fully opened-- just 3 days! She commented on this, and started letting them bloom more.
The zinnias remind me of my last suggestion: redirect them to something they ca do with abandon. Let calendula or another flower reseed and grow abundantly and unleash your kids on those. Let them pick bundles of Spanish Bluebell and dandelion until those plants are just about dead. Say "you do what you want with this one" and "that one you can carefully pick a couple but leave the rest" and still others "this one is special and I'd like to leave it there"" and show them how they can interact with it without destroying it (smelling, touching, watching the bugs on, etc.)
In other words, demonstrate ways they can interact with the garden beyond focusing on pulling out plants. Then, wait it out. Keep reminding them to not pull plants out (which is minimized because you've curtailed adding in new plants in the same way you keep something breakable away from them until they can internalize your wishes). And wait. There will come a day when they will stop.