VI component tester not working properly when capacitor is placed parallel with diode
So how should the tester know you're testing the diode but not the capacitor? 

Remember what inductivity and capacity do: they introduce a lag between current and voltage (or vice versa). 

So, no, if you have a capacitor in parallel to your device under test, you can't get a correct V/I curve for that device if you excite it with an alternating current.

What you can do is do a really slow increase in current, wait until the steady, DC case has settled in, and then move on to the next higher current.