How to recover data from iPhone backup files WITHOUT the info.plist, status.plist, manifest.plist, manifest.mbdb, and manifest.mbdx files? I have iTunes 10.5 and Mac OS X version 10.6.8.

I would like to restore my iPhone from backup.

iTunes detects all my backups EXCEPT one, which happens to be the backup I want. 

I looked in the backup directory (~/library/application support/mobilesync/backup) and IT IS THERE along with the other backups. HOWEVER, unlike the other backups, it's missing five files: info.plist, status.plist, manifest.plist, manifest.mbdb, and manifest.mbdx. 

I have tried applications that can extract data, like the free iPhone Backup Extractor for example, but they work only if I have the five files. 

Is there a way to recover data without the five files?
It depends whether the remaining files you have are from an encrypted backup.

It won't be feasible to read your backup data if you have an encrypted backup with no Info.plist, Status.plist, or Manifest.mbdx files 

However, if your backup wasn't encrypted, it will be possible to read that data. You'll find that whilst the filenames are unhelpful (they're SHA-1 hashes of the file's internal path), the file contents easily readable. It's only the filenames that'll be lost.

If you're looking for images, try renaming a bunch of the files to have .JPG extensions, and see whether Windows Explorer or macOS Finder previews them for you.

There are a few approaches for digging into what's there in bulk. You mentioned using iPhone Backup Extractor, and there's a guide on how a use "Bulk Rename Utility" to read backup data when in this situation.