What you are describing suggests that your knee extensors, and particularly the single-joint quadriceps muscles—vastus lateralis, vastus medialis, and vastus intermedius—are weak. The closer that you are too the floor, the greater the moment produced at the knee. At 0°—that is, parallel to the floor—the moment is at a maximum due to the weights of the upper and lower extremities acting perpendicularly to their alignment. Thus, it is at this time that the knee extensors must work the hardest. And it is also, consequently, when they are most likely to fail.
To answer your question directly, it is not something that you are doing incorrectly, but rather a weakness in the chain. And doing push-ups on your knees will not help your development in this regard because that will simply circumnavigate the problem rather than addressing it.
A better choice is to perform your push-ups on an adjustable incline, such that you can progress gradually toward parallel. This ensures that your posture and muscular recruitment are unchanged, and that only the load is altered. If you have access to a Smith Machine, the bar provides a perfect adjustable platform from which you can perform your push-ups. (The lower you set the bar, the more closely the exercise approximates a push-up from the floor.) Otherwise, a low wall, stool, or similar stable platform will suffice, albeit with limited variability.
If you do not have access to equipment, you can still progress with careful attention to your position and posture. The muscles will develop with time and effort.
Attention should be given to bracing the abdominals and forcing the legs out straight, such that the body is rigid and immovable from head to toe throughout the movement. This is particularly important as you approach parallel.
And finally, you might also benefit from supplementary exercises that target the knee extensors.
I hope that helps.