I've been working on one that runs in Firefox[1], but there are many others that look promising for Chrome as well.
[1] https://addons.mozilla.org/en-US/firefox/addon/brook-feed-re...
I use a web interface, The Old Reader, to access feeds from 4 devices. Whichever I'm using at the time knows which articles have been marked as read; without using a different client for each OS platform.
So a browser extension that syncs to one's Firefox account, like the bookmarks/history feature would be handy, with no additional hosted service other than having a browser login.
Personally, I find that I read very different things on my laptop than I do on my phone (technical articles vs. news) so I haven't had a huge motivation to investigate it, but it's an interesting avenue. The only possible stumbling block is that the size of sync'd data is relatively constrained, about 200k I think, which is probably enough to hold a user's feed list and track whether they've viewed/ignored various articles, but it would require some planning.
There are many parsers that are adequate, especially if you don't try to get too fancy. If you're curious, the source is available on github. https://github.com/adamsanderson/brook