pushing your elbows up gives you tendonitis? that.. doesn't make sense. Tendonitis happens over a very long period and is basically a lot of inflammation gathered up at the tendon from overuse. I don't get how you would get tendonitis in the elbow from.. front squats. There is practically almost zero pressure at the elbow joint.
I personally find it better though I didn't suggest this. I hope you aren't misunderstanding pushing the bar with your palm towards your throat.
Also I don't see much point in going just parallel or doing Box(or bench in this case) front squats.
Set the back tight, keep the barbell where it should rest and come up. Sometimes the upper back will still buckle when going maximum or near failure. You can help this by doing less weight and focusing on it specifically to strengthen it.