It sounds like gardener's hell, and it's not going to be an easy job. What you have to do is kill the root, not just the growth. That means you have to track back each vine (cutting as you go,obviously) and cut it off at the soil. Then you have a choice - dig up the root if it's small, or hit the stump with undiluted Round-Up concentrate.

Whether you dig or use RU, you're going to have to keep a weather eye on the whole area for a couple of seasons, as some vines grow from an underground (brain fart - can't remember the exact term) root mass - bindweed and passionflower are two that grow like this; non-vines are Canada thistle and mugwort.

Ken Adrian, who often posts in the Shrubs and Trees forums, has great instructions for using Round-up for jobs like this; I'll track that down and post a link under this.

A picture would really help with more advice, as you may have some vines that are easier to get rid of than others.

Here's a discussion from the Trees forum in which Ken, who is a very funny man, gives his Round-Up instructions; it's in Ken's post from 6/21 and is about half-way down the page.

I hope this helps. As I said before, it sounds like a big job and I wish you lots of patience and all success with it.

Here is a link that might be useful: Look for Ken's 6/21 post

Ok so here are some pics to take a look at. The first one shows the overall general state of the area I'm clearing. I know there are a few small mulberry's under the vines and a few other small trash trees. Obviously digging up each vine doesn't seem to be much of a possibility, it would probably take me a year of steady work to go that route.

hmm image seemed to disappear from the post somehow.

Wow. You are a good person to take that on for your parents.

I still think that you've got to cut it down and get at the roots, but jeez. If I were facing that, I would pick an edge, mow or weedeat that stuff down a couple of feet across the whole side, then dig or Round-Up the "stumps", then repeat. And repeat. And repeat. Maybe immediately mulch what I've cleared,to try to minimize seeds sprouting as I finish.

I can sure understand your wish for a flame-thrower.

I don't recognize either of the vines; if you don't get IDs here in a day or so, try posting this in the Name That Plant forum, and you'll get IDs and advice pretty quickly.

The vine with the dark, heart-shaped leaves against the brick is Dioscorea bulbifera aka air potato - a HORRIBLE weed.

The other one is a wild morning glory, but won't be able to i.d. it until you post bloom pictures.