George Porter's spectacular goal looked to have won it for the hosts, but Preston popped up to head in the superb Ryan Sellers' cross in added time.
Josh Staunton made a welcome return from injury off the bench with an equaliser to cancel out Frankie Sutherland's first-half opener after his penalty had been saved.
Town could count themselves unfortunate to be behind when Sutherland smashed home the rebound after his penalty had been tipped onto the post superbly by Sam Johnson.
Dayle Southwell had a chance to level but could only divert Ryan Sellers' drilled cross over, while Jordan Preston had earlier seen his deflected cross hit the top of the bar.
Another drilled cross by Sellers, after another good run by the full-back, was nodded over by Jonathan Edwards.
But, from Simon Lenighan's hopeful pass forward, Bromley needlessly conceded a corner, and from that, a floated cross from the right was met with a cushioned header by Staunton at the back post.
Porter had a free header with 15 minutes to go, but his tame effort was straight at Johnson.
Town were then on the ropes as Sellers cleared off the line from Uzokwe, before Johnson spilled a header back across goal, requiring a scrambled clearance.
Bromley were in the ascendancy now, and their pressure told when Sutherland's magnificent pass behind the Town defence played in Porter, who produced a stunning volley across goal.
But Preston's late leveller earned a share of the spoils.
Harrogate returned to winning ways as they fought back to beat Boreham Wood at Meadow Park.
After two games without a victory - or a goal - the visitors fell behind when Chesterfield loanee Gozie Ugwu nodded home.
But Harrogate levelled eight minutes later when George Thomson headed a cross from Joe Leesley back across goal and past the keeper.
The Yorkshire side then took control with two quickfire goals at the start of the second half.
Wood goalkeeper Ryan Huddart's clearance hit advancing forward Aaron Williams in the back and flew into the net, and two minutes later Leesley's cross hit Femi Ilesanmi on the knee and crossed the line.
Ugwu reduced the deficit with a 30-yard screamer, but Jack Emmett wrapped up the points for Harrogate when he curled in a fine long-range effort.