Using rar you can list the contents of the available parts, and extract complete files, from a partial set of parts of a multipart archive. So, if you have file.part01.rar through to file.part03.rar in a multipart archive with 15 parts, that is the last part would be file.part15.rar, you can view what files are in parts 01-03 using the l command on each available part. You can extract, using the e or x commands, any file that is complete inside parts 01-03.

Note, that in all cases you must have the parts for the multipart archive starting with 01 and those contiguous after. That is, if you have parts 01-03, rar will list what is inside each of the parts you name. If you test the archive, using the t command, rar will test and list the files in each part that is contiguous from 01 until the first missing part with the lowest sequence number. in the example that would be parts 01-03. Rar then displays the error:
Cannot find volume <filename>.part<n>.rar
meaning, in the example where <filename> is file and <n> is 04, it cannot find part 04 and then displays a checksum error on the last file in part 03, if it is incomplete.

If you have parts 01-03 and parts 05-15, rar can only list what is in parts 01-03. If you test the archive rar will test and list what is in parts 01-03 and then display the above error saying it cannot find part 04 and then display a checksum error on the last file in part 03, if it is incomplete. The rar test command will ignore parts 05-15 because it cannot find part 04.

Prior to bug fix 245703, when Ark opened a partial rar multipart archive, Ark displayed the error message:
Loading the archive /<path-to-file>/<filename>.part<n>.rar failed with the following error: Extraction failed because of an unexpected error.
because the multipart archive had only some contiguous parts, starting with 01, available, clicking OK would list, in the infobox, the available files inside the available parts from part 01 to the last contiguous part available. In the example that would be parts 01-03 of the archive, whether or not any parts after part 04 were available. From the file list in the infobox you could extract any of the files that it listed, the same as rar can.

Bug fix 245703 causes Ark to display the same error message because only some parts of the multipart archive are available. Then whether you click OK or Cancel, Ark clears the infobox. So Ark does not list what is in the parts of the archive that are available, even though rar can. Also, because Ark clears the infobox you can not extract files inside archive parts that are available, even though rar can.

Reproducible: Always

Steps to Reproduce:
1. Create multipart rar archive.
2. Move one or more parts from the archive directory.
3. Open with part01 or any part with a sequence number less than the first missing part of the archive Ark.
4. Reply OK to error message about not finding a part of the archive.

Actual Results:  
Infobox is cleared and no further action of archive not possible.

Expected Results:  
Infobox lists files in contiguous parts of archive, starting at 01, up to the first missing part with the lowest sequence number. This allows for extraction of files inside the list in the infobox.

Note that rar list only lists single archive parts, whereas Ark listed all contiguous parts of the archive, starting at 01, up to the first missing part with the lowest sequence number  

I have not tested rar functions other than list, test and extract. I have not checked or tested archive types apart from rar.