I'd like to experiment with making purées (particularly carrot and parsnip, if it matters), but I don't have a food processor or any other mechanical mixing/blending tool. Is there anything I can do or is a processor necessary?
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).
You're just not going to get the same texture with anything that doesn't have spinning knives (blender, food processor, etc.) You might be able to get close with enough cooking + a masher, then a hand mixer (or do the old hand-crank ones still count as mechanical?).
As things go for the type of cooking you're dealing with, consider a stick blender -- they're much cheaper than a food processor, don't have the issue with leaking seals as a bar blender, and they don't take up as much storage space.
They can't quite do the same things as the others (bar blenders are good for breaking down ice cubes for smoothies that a stick blender might choke on, and you can leave them alone while they work; food processors can shread or slice with the right disks, kneed dough with the plastic blade, or break down solid objects with pulsing to keep it from getting to pureed), but I don't think the others work for making whipped cream, either.
A sieve and forcing through the food with a spoon will work.
Thoroughly-cooked carrots are at least as soft as boiled potatoes, so you should be able to get a good result with a potato masher if you are patient. Use ample warm liquid (water, stock, ...) to thin it from mash to purée as you go along, and a generous knob of butter wouldn't hurt if your recipe will accommodate it. Once the purée is thin enough, a bit of brisk work with a wire whisk should take care of any little irregularities.
If you are VERY patient and completely bereft of tools, you could probably even do the job with a fork, but that strikes me as verging on masochism.