Quicken equivalent

You might want to try gnucash.

sudo apt-get install gnucash

It's a good all around personal finance manager. I think it can even open Quicken files (not sure though!)

Homebank

«HomeBank» is free software. Use it to manage your personal accounts. It is designed to easy to use. Analyse your finances in detail using powerful filtering tools and graphs.

Have you thought of taking it to the cloud? I believe Quickens online version has been replaced with Mint. That will free you from platform dependencies altogether. I haven't used it myself as its really targeted at US users and I am not in the US, so I can't say if it does everything your parents would need, but its worth a look.

Regarding Linux native apps, I have been looking for one myself for quite some time and just couldn't find one that would do what I needed. Financial and accounting apps in Linux are an area in bad need of improvements IMO. I finally went to the cloud with Kashoo, but i don't think Kashoo will do what you need hence the Mint reco