Parsnip cosmetics

It's often carrot root fly damage.

If it's canker, it can be down to drought, poor drainage, over rich soil, or if damaged by root flys etc., try growing canker resistant varietys such as gladiator, add a little lime if the soil is a bit acid and double dig to improve drainage.

Totty

I'd agree with Totty on a number of things, however, whether it is canker or just physical injury like damage (both are possible, from what you write I'm tempted for the latter) both are worsened and can be simply induced by the size you are proud of.  I'm not joking with you.

I'd recommend Avonresister, if it were canker, I'm uncertain though it is.

Otherwise, for size and physical damage, try to keep even moisture levels.

I also do not protect against carrot root fly on parsnips. And indeed there are less damages than to carrots.

And indeed any disease can get in through any injury, let that be from insects or other inflicted damage.

I do not get canker. Apart from Avon, I usually grow Hollow Crown (till May sowing). And back on topic, I wrote earlier, the professional I trust says later sowing, smaller roots, less problem.