I was close, it was '74 when Firestone quit making them.
When NASCAR is born in 1947, the race tire market is solely dominated by the Firestone Tire Company.
NASCAR's emerging popularity in 1954 gets noticed by Goodyear and they re-enter the competition by using police car tires at the now 4-year old Darlington speedway, and finally defeats Firestone five years later establishing a new speed record with Jim Reed at this same track. 1966 is a milestone year for the racing tire industry when Firestone and Goodyear started manufacturing the "Inner Liner Safety Spare", called Lifeguard for NASCAR racing.
This tire is in fact a second envelope inside the tire, which prevents instantaneous deflating and allows the driver to return to his pit after a flat. From now on, all NASCAR are be equipped with the Lifeguard.
The two tire giants will alternate victories and speed records when Firestone stops producing racing tires in 1974. Goodyear remains the sole supplier until 1987, when a company from Indiana, Hoosier, starts supplying racing tires, first in the NASCAR Busch Grand National series, then in Winston Cup racing, where Hoosier wins several races.
It is interesting to notice that all tires manufactured so far are bias-ply type, when in Europe Michelin had long ago invented and patented the radial tire. The story doesn't tell if Michelin's patent expired, was bought or turned around, but in 1990, Goodyear proposes the Eagle radial race tire which immediately proves considerably superior to the now obsolete bias-ply tire. This puts a temporary end to Hoosier's contribution in NASCAR Winston Cup. The following years see new generations of tires and improvements of the inner liner.
Hoosier tries returning to Winston Cup but succeeds to win only 3 races during the 1994 season and withdraws from Winston Cup and Busch Grand National, to concentrate on the smaller series, such as Winston West and Featherlite.
Today, Goodyear's yellow-lettered EAGLE tire is the only one used in NASCAR.