Parsnips are very poor germinators at the best of times and their
germination rate diminishes quickly, the longer they are kept. It is not
advisable to transplant parsnips, better to sow a few in situ and pull out
any extras that you get in the same place. I found like you that beetroot
do
well from transplanting but I think transplanting parsnips is pushing it a
bit, but, you never know, they have two choices if you want to try it

I took over an allotment in July, and amongst the weeds there were several
established parsnips that the previous owner had obviously planted in
spring. The leaves on some were over a foot long, but they had never been
thinned and were growing in clumps of 5 or 6.

I broke the clumps up, and transplanted the whole plants with their skinny
roots of several inches long, some into another bed.

All of them survived the transplanting, and I have had parsnips over two
pounds in weight off them recently, and I have a few left in for christmas.

The only thing to note is that it is pretty difficult to plant a long root
straight. I got very forked parsnips in some cases.