A friend asked me about this and the Sporting News record book lists
the record as 14 in three different extra-inning games.

However, my friend was asking if the 9/1/82 game between St. Louis and
L.A. in L.A. was one of the three and it wasn't listed.

According to the Retrosheet boxscore
http://www.retrosheet.org/boxesetc/B09010LAN1982.htm
I count 14 pinch hitters:

St. Louis
Braun for Tenace
Iorg for Sutter
Paris for Bair
Herr for Oberkfell
Hendrick for Landrum

Los Angeles
Roenicke for Russell
Brock for Yeager
Mota for Scioscia
Monday for Belanger
D. Sax for Taveras
Orta for Reuss
Morales for Beckwith
Marshall for Howe
Maldonado for Niedenfuer

Bob Timmermann