Or... something else that looks like a good possibility is "Belle Coquette" but I can't find much information on it. (http://www.helpmefind.com/rose/pl.php?n=50265)

One more to consider...

MRS. DOREEN PIKE http://www.heirloomroses.com/cgi/browse.cgi?page=item&cat=29&item=77

'Lilian Austin' was introduced in the 1970s and 'Belle Coquette' and 'Mrs. Doreen Pike' were introduced in the 1990s. They would be nice to grow if they remind you of your parents' rose bush.

A wild guess is that the original might have been the rosa blanda hybrid 'Betty Bland' which was introduced in the 1920s. It has small fragrant pink double blooms and makes a 6ft bush that survives harsh Canadian winters. Check the description below and click the photos link to see if the pictures match your memory:

Here is a link that might be useful: Betty Bland

I think you may have it! :)

And what's more, I remembered the other day that there is a large bush of this rose in a nearby cemetery - so I'm planning a trip there this summer to get some cuttings and to look for baby bushes underneath or nearby.

My mom said she managed to transplant a bit of her bush by getting some of the root, but she never tried to do cuttings.

Getting cuttings from the original plant or transplanting pieces with the roots sounds like a fine idea. I wish I had done the same with mystery roses at my grandparents' home -- they are long gone.

Let us know how it goes this summer -- best of luck!