I know it's not the same, but I *think* celery root (celeriac) is similar to parsley root (I'm sure someone will correct me if I'm wrong). I start my celeriac seeds indoors in February, and transplant them mid-May for a fall harvest. Worth a try if you've been unsuccessful so far?

celeriac root is different, as far as I know it is round with several long roots attached(at least it is what my mother grows, it called apple celeriac). So couple of extra roots I guess wouldn't change much on celeriac root. Root parsley looks more like carrot, and spited root is not good, same as for carrots. It is night mare to clean it when you have a root 1 X 1 inch with 4-5 attached long "tails" less then 1/4 in diameter. I think I am going to try, but with some preparation. I am making 5 inch tall 1.25 inch wide tubes from brown craft paper. They fit snugly into standard plastic cells for the trays. I will fill them with mix, start parsley, and when it is time to go in the ground will plant them with paper tube on them - hope this way the plants will have enough length for their root and roots will not be disturbed during transplanting. This is a plan) Will see how it goes.

I guess I was thinking because I heard the two will cross-pollinate that they were similar. But thinking about it, you're probably right because parsley root does resemble a carrot more than celeriac. Sounds like you have a good plan. Good luck & let us know how it works--always looking for new things to try...