Where does Word keep its pinned documents list?

It isn't in a file, so there is no file path. Word stores that information in the registry, in the values under the key HKEY_CURRENT_USER\Software\Microsoft\Office\14.0\Word\File MRU.

The key contains up to 50 values with names "Item 1", "Item 2", etc. Each value contains a string like

[F00000000][T01CFA783D052DEC0][O00000000]*C:\test1\test.docx

The file name, of course, is the part after the asterisk. The first part is [F00000000] if the document is not pinned on the list, and it is [F00000001] if it is pinned.

Frankly, I wouldn't mess with this. Just use Word 2013 to open the documents that you want to have pinned, pin them there, and be happy.

I suppose if you had a ton of pinned documents, you could export the Word 2010 key from the registry to a .reg file (which is just a text file you can open in Notepad), and edit the file to remove all the unpinned entries. You would also have to change the key's name in the .reg file to the correct value for Word 2013. That file could be imported back into the registry.

However, there's another wrinkle: Word 2013 doesn't use the key HKEY_CURRENT_USER\Software\Microsoft\Office\15.0\Word\File MRU to store the information, although that key does exist. Instead, because it allows multiple accounts, Word 2013 looks at the key HKEY_CURRENT_USER\Software\Microsoft\Office\15.0\Word\User MRU, under which there is a separate key for each account, and each of those account keys contains a File MRU key and a Place MRU key (the latter for the Recent Places list).

So you have to look at the registry on your Office 2013 installation, and edit the .reg file to refer to the File MRU key under the correct account. To know which is the correct account, if there are two or more, start Word and create/save a document with a distinctive name. Then look at the registry to see where that document's MRU entry appears.