Okay, so, looking at the examples in your flickr gallery, including the ones from your parents' point and shoot camera, I see a number of things:

1. You're seeing the effect of a smaller depth of field, which is more apparent with a larger sensor size. In this example, you are shooting at f/5.6 and 26mm, and it looks like you're pretty close to your subjects. You're focusing on the little girl's face, or maybe even a bit closer, and everything away from that plane naturally falls out of focus. For more depth of field, stop down more. You can also probably get better results by focusing a little further back into the picture, so everything is more in focus on average.

2. I think you're looking too critically at your new photos and not critically enough at the examples from your parents' camera. First, yours has a lot higher resolution, so when you zoom in all the way, you're zooming in a lot more. Second, the point and shoot camera is applying a lot of noise reduction and sharpening in-camera, and the results look fine at first glance but actually have lost a huge amount of detail. I certainly wouldn't say they look better than what you are getting. If you make normal-sized prints (8×10, say), they all should look fine, and similarly they look fine at web viewing resolution.

3. Your focus is a little off in some of them. But again, I don't think it's a deal-breaker. Sharpness is overrated. To improve, though, try to be a little more careful with where you put the focus points, and make sure to lock focus and avoid recomposing after that when you are close up — better to put an alternate focus point directly on the subject's eye if you can do it. Again, this is more important because of the higher resolution and shallower depth of field of the more capable camera; the point and shoot isn't any better at this, but you can't see the difference as well.

4. All that said, do be aware that you are using a budget, entry-level lens. It won't give you miracles, and while these class of lenses are generally decent performers, if you put them to "stress tests" and look for defects, you'll never be happy.

And kind of unrelated but it's hard not to say something: whatever post-processing you are doing to the eyes in this one seems really strong, crossing over into the uncanny valley for me. Your taste may differ, but I'd suggest toning it down.