I've been using Windows Access since 1993 and Excel since 1990 (Mac) and 1993 (Windows). There's no real substitute on the Mac platform for Access, although Excel's file size has allowed deeper/larger data sources for a few years. I've managed to use Excel to provide a data resource for products that Access was the only option until a few years ago. I tried Filemaker but found it was a pretty weak option for larger data pools.
I've been "all in" on the Mac platform for over a decade and wished for a competent alternative for Access - UNIX is so much more powerful than anything on Windows - but I'm still using Access. Sigh.
My solution was to sign up for Office 365 Home and Office, so I get Access in that package. I use Parallels 10 and Windows 8.1, with Access 2013 installed on that partition. I bought a year's subscription to Office 365 via Amazon - a 40% discount over MS's cost.
I get Access 2013 for about $3 per month after the cost of Parallels ($50 upgrade) and Win 8.1 ($40). And I don't have to ask for the "Mac equivalent question" you did.
I also won't use Excel 2011 as a substitute for Excel 2013. I've been using Excel since version 4 in college. Excel 2013, and its database size may be what you're looking for. MS's newer iterations may be nice to use, but Excel 2013's file size (huge) may serve what you're looking for - it's working for me and my small company. Excel 2011 is too gimped and there's nothing I can think of on the Mac platform that can compete with MS's offerings on Windows.
I just found this thread, and I have the same issue/challenge.
I recently tried your approach. Office365 won't allow me to download/install Access via Parallels. Are you still able to run under your setup? i.e. I was wondering if MS changed something to disallow it.