Lewis McGugan's early goal of the season contender gave Forest a 15th-minute lead and Bolton's haphazard approach to possession allowed them to dominate the opening half hour.
Chris Eagles scored his second goal in four days to level after Kevin Davies' header came back off the post and, when the veteran striker pounced on an error from Greg Halford, strike partner Marvin Sordell scored his first Bolton goal four minutes into the second half.
But Reid brought Forest back on terms with a clinically-taken equaliser shortly before the hour and despite the best efforts of both teams, a winning goal was not forthcoming.
O'Driscoll can look with satisfaction upon an opening of four league and cup games without defeat and, although opposite number Owen Coyle might have hoped for a better return than four points from the first three npower Championship matches, Bolton have themselves recovered well from a chastening opening loss at Burnley six days ago.
Done no favours by a playing surface subjected to a pre-match downpour, Bolton skipper Davies was booked for a meaty challenge on Forest full-back Dan Harding inside the opening seconds.
The visitors started the brighter of two unchanged sides as McGugan, Simon Cox and Dexter Blackstock all caused problems for the Bolton defence, meaning due warning had been served when the opening goal arrived in spectacular fashion.
Blackstock touched the ball infield to McGugan, who let it roll across his body before unleashing an unstoppable 30-yard drive past Wanderers goalkeeper Adam Bogdan via the underside of the bar.
McGugan's next contribution was almost as sumptuous - a defence-splitting pass played with the outside of his right foot that sent Cox into the area, only for Mark Davies' last-ditch challenge to force a corner.
Bolton belatedly made inroads, but Sordell and Kevin Davies were both well marshalled by Halford when attacking crosses from the left.
The equaliser arrived from the same source six minutes before half-time as Kevin Davies nodded a cross from full-back Sam Ricketts against the post and Eagles drilled in the rebound.
Having appeared close to disarray moments earlier, Bolton had Forest in plenty of trouble before the interval.
Reds goalkeeper Lee Camp was at full stretch to keep out Sordell's header from another Ricketts centre and Halford had to hack out of the six yard box after Tyrone Mears was released down the right.
Halford's first contribution of the second half was far less beneficial for his side as he hesitated in dealing with Ricketts' speculative ball forward, allowing Davies to steal in a tee-up Sordell to prod home off the post.
Sordell might have extended the lead when he blazed Eagles' cut-back over and Forest appeared to be on the ropes when Camp misjudged a 55th-minute corner from the former Burnley playmaker and a goalmouth scramble ensured.
Moments later, for the second time in the match, an equaliser arrived against the run of play.
Blackstock knocked down a long ball to Cox and his smart lay-off allowed Reid to crisply find the corner.
Befitting of the game's thrilling end-to-end nature, Mark Davies almost restored Bolton's lead immediately, but his diving header from Eagles' left-wing cross flashed past the near post.
Namesake Kevin was the next Wanderers player to waste the chance of a winner, cushioning a volley over when left unmarked from Lee Chung-Yong's corner.
A set-piece at the other end almost handed Forest the points, but Mears found a vital touch as Cox stood was poised to convert Halford's flick-on.