When I go through the exercise of analyzing not only the pictures I have taken, but also pictures of some other photographers, over and over again some of my favorites are very "simple" images which go in the opposite direction of the idea of "show it all". They depict a single idea or object. There are no distracting elements. Your eyes settle immediately on the subject and they are all yours to admire. I feel this is sort of like going to a museum and having a wall with all these different paintings competing for your attention versus having just one hanging there.
We talked before about getting close to our kids faces and focusing on their eyes, but don't just stop there. Look elsewhere also: a favorite toy, what they are wearing, what they are doing with their hands, a facial expression. Concentrate on those unique things that remind you of your children, your family, a good time spent together.
The good news is that all you really need is your existing camera and the knowledge that you, as a parent, know what matters. A basic zoom lens can help you get closer to your subject but remember that you can always crop the picture later if you have too. You can try including some vignetting in your picture to help isolate the subject in the middle of the frame.
Particularly with children, it seems that their hands are always trying to reach something new to explore. That's a perfect time to take pictures of their hands manipulating a crayon, a toy, etc.
When you look around and search for all those "details", a whole new photographing opportunity opens up for you.