I have a friend who wants register (for example) www.mysite.com
for his English content

AND

www.mysite.kr
for his sites Koren content.

Is it better for him to set up www.mysite.kr as a parked domain
or as an add-on domain of the main site.

I don`t quite understand how parked domains and addon domains handle sub-directories. Would the URL betray the fact that these are not "genuinely" separate domains?

Parked and with 301 redirect. But first you need to choose main domain name for your web site.

If the .kr site is going to be Korean content, parked and redirected is ABSOLUTELY NOT the right thing.

For a static site, out of the two choices only an "Add-on Domain" would be the one to use. This will keep the two sites as separate entities.

However IF you are using a CMS that is capable of delivering different content (languages) based on the requested hostname then parking the .kr on the .com will be the way to go.

Unless the main site links to the subdirectory that the addon domain is mapped to, no user agent will ever know it is there. And because the root of the addon is the parent subdirectory, no user agent can go UP the directory tree into the parent site structure.

So as an "Add-on" domain, would it still be able to share resources with the "main" domain, such as images, style sheets, javascript files and so on?

Not in the way I assume you are considering.

You could use the ../ syntax to go above the site root if "parent paths" are enabled on your server.
Other than that use absolute addressing as if they were remote files, because it is the same server speed implications are pretty much non-existent.