To my knowledge, no formal, as in Barnes company list of serial numbers, currently exists. As a company, Barnes did keep very detailed records. Know a very limited list of serial numbers with the dates made exist for some of their wood working machines like the #2 scroll. That came from somebody that got some of the records from the Barnes factory when it was shut down many, many years ago.
Barnes did not take any of the patent dates of the castings to my knowledge so that would not help. For those that are into "the details of Barnes", studying old catalogs can give some indications. There is a book called American Foot Power and Hand Power Machinery by Ken Cope. It includes almost a page on the #13 as a copy from an early 1900's catalog. It says the #13 was introduced in 1901 as basically an enhancement to the #6 metal lathe and was priced from $162 to $212 based on bed length.
There used to be a book of serial numbers for many machine tool builders. It wasn't perfect but possibly has the dates you want. Had a pretty obvious name like "Machine Tool Serial Numbers" ... dealers used to buy it.