Can you purée without a food processor?
I think some sort of mechanical aid to macerate the food is necessary for your project. Depending on what dish you attempt, there are few options. 

A ricer (or food mill, depending on who you buy from) is a manual device (non-electric) that you might run cooked potatoes or parsnips through. Many people do mashed potatoes this way. Depending how cooked your parsnips get, a hand blender (or immersion blender, stick blender) may work. 

However, all of this requires you to shell out some cash for a device. I'm not sure, from your post, if that's an option. If you're going for  ultra-smooth texture for a bisque or other soup, it might be worth it to get one of the small-bowl food processors (just work in batches). 

Good luck!