I did quite a number of zero gravity manoeuvres on a Cessna Citation II, not as a pilot but as an experiment coordinator.
The engine lubrication was gravity based on that aircraft, and zero-g caused a Master Caution / oil pressure warning. Since the zero gravity manoeuvres lasted only for about 12 seconds, this wasn't a problem according to the engine manufacturer.
Entry and recovery caused extra g-forces on the airframe, I remember we reached 2.5g during pull up. Especially the recovery out of the parabolic flight was tricky, balancing between duration of the zero-g segment, g-force during pull up and overspeed. Sometimes spoilers were deployed at the end of the parabolic section to keep the speed down.
There isn't much risk of stall, since the higher wing loading occurs at quite a high speed, keeping the required lift coefficient low. The low speed segment on top of the parabola is a zero g, so no stalling there.
Another serious risk is getting hit by things flying around the cockpit and cabin when negative G is accidentally achieved. Flight manuals crashing into your head at 2.5g during the pull up are no fun.
Negative G's are also a risk for the fuel system when it is purely gravity based. If air bubbles get trapped inside the fuel line it can cause engine stall. I think this is less of a problem in jet / turbine engines which seem to have fuel pumps in any case.
On the Citation II negative G s caused a spray of fuel from the vents in the wing tip; spectacular but harmless.