Try some vinegar. It's a potent herbicide, especially if sprayed on a sunny day. Non selective, so don't get it on the plants you do care about.
If the parsnips are few and far between, you can chop them at ground level and put a spoonful of salt on the root. This method has been the death of many giant thistles at my place and it works on anything with a big taproot that tries to resprout.
Let them start to flower and then chop the flower spikes off, they flower and then die so do that a few times and you will get them. mowing takes several attempts but will do the same thing and kill them in the end.
I have a horrible "hay" grass that is invading my garden spot from the fields nearby for many years before I came along. Spreads by rhizomatous roots which can be as deep as 4 inches or more and still put up shoots because there is a green leaf somewhere feeding the whole complex mat!
Lasagna didn't work from last year because I didn't fully understand this habit. I did the best I could this year before planting potatoes and onions, but it's still coming up in there.
Will vinegar kill this? before it kills me!!
Innovations that are guided by smallholder farmers, adapted to local circumstances, and sustainable for the economy and environment will be necessary to ensure food security in the future. Bill Gates
Not looking good. I think this might be the end. Wait! Is that a tiny ad?