You almost got it, when a yoshi is using parry to counter moves you shield, invincibility frame active, *here is where you're wrong with your sequence for parry > nair* just jump before parry frames end (not a double jump cancel) now you have the choice to do a rising nair or a double jump canceled (djc) nair, for your example yoshi is fighting fox, if a fox nairs you they usually land with a shine in case you shield but if you do a djc nair out of parry the shine will catch you however if you do a rising nair you will hit before the shine, the benefits to the djc nair is you land sooner allowing you to follow up after the nair. Yoshi usually parry into djc nair because as a enemy hits you they often drift through you, the double jump lets you remain mobile enough to stay close enough to the foe for the nair to connect.
Other options out of parry are djc up-airs, these are harder to hit out of parry but not that much more difficult and depending on percents it can lead to a much better punish than what nair would. The other option you have is to parry > double jump cancel land (if you do this yoshi does not leave the ground but will have canceled his shielding animation and you're free to act just like you would be out of a no impact landing) this gives you the most time to punish, this will let you land a tilt or d-smash
Another thing that most people do not realize about parry or even powershielding is that if you crouch before the attempted parry/powershield when you enter the shielding animation you stand up moving you into the approaching attack allowing you to make the timing easier
