I don’t think Twitter directly provides anything, but there are a handful of services out there that can generate a feed from Twitter for you (try rss.app. The EE RSS Parser shouldn’t have any trouble parsing the feed once you have it generated.
Other options of interest or for those who might find this post later on:
I do remember that some years back Twitter actually did provided RSS feeds directly. Not sure why they removed that option.
My question is more about the RSS parser on EE. Assuming I turn a tweet or anything else into an RSS I could be able to format that content directly on a template page correctly using this addon?
Yes you can create an RSS feed and display using the RSS Parser plugin. Your feed can contain any entries you want.
I do just this for a company that has their main site and a bunch of EE powered micro sites. I use an RSS feed to push out things like news and testimonials to the micro sites, so they publish once on the main site, then items such as filtered news headlines and pulled into each micro site as required.
Addendum: I actually use several feeds for various purposes.