Don't get too hung up on which distribution to install - it is the applications which really matter, and the main applications are the desktop and the web browser and the word processor. I think you would be best off getting a distribution that has a wide choice of desktops and checking them out one by one; try XFCE, LXDE, Openbox etc. You can also try Gnome: your parents may prefer a richer desktop that is a bit slow to a faster but more basic one. Similarly you could try out the various word processors that are part of your chosen distribution. (I'm assuming that you will have to sort everything out, and make a judgement between how fancy / how quick the machine is before handing the machine back to them.)
(As an example, I am running OpenSuse 11.2 on an old Dell CPx laptop, 500MHz processor, 192MB of RAM. OpenSuse is not normally considered lightweight but during the install I just chose XFCE rather than KDE as the desktop. With Google Chromium running with 5 tabs open, and a terminal with htop running, htop is reporting that I am using 99MB of memory.)
You may need to be careful how you install Linux: your computer will find it much easier to boot an installed distribution than a live CD. On this machine I installed from the network install CD without a problem, for Ubuntu based systems you would be better off choosing install straight from the boot menu (assuming Ubuntu hasn't changed that much from what I remember).
As for remote assistance for your parents, if you are happy using the command line, then I would just run the ssh server of your distribution. You will probably have some nice pointy and clicky way to set what services start on whatever distribution you use, but the method will depend on the distribution; come back and ask if you can't find how to do it.