John Still’s side dominated proceedings in the early stages of the game with the first chance falling to Jack Taylor in the opening minute but his volley from close range was saved at the near post.
The hosts had another chance to break the dead-lock soon after when Coulthirst’s shot-on-the-turn from the edge of the area fell wide.
It was Walker’s turn to pose a threat on six minutes when he was played through on goal but his low-placed drive fizzed wide of the post.
The Chances kept falling to Barnet in the opening 10 minutes and Charlee Adams would be disappointed his ambitious effort from distance flew narrowly past the goal-frame.
However, The Bees did have their noses in front on 11 minutes when Coulthirst buried the ball low and hard from the spot after the referee awarded a penalty following a push on Dan Sweeney.
The opener prompted the visitors to tighten up their defence and the Bees found it hard to maintain their earlier superiority.
However, after a corner was cleared to Craig Robson, the temporary skipper hit the ball first time from close range but Ben Dudzinski did well to prevent the hosts from doubling their lead.
Hawks skipper Brian Stock found his name in the referee’s book in the latter stages of the first period after a late tackle on Adams from inside the centre circle.
Nicke Kabamba posed a threat on the stroke of half-time when his cross deflected onto the roof of the net but The Bees held on to take a one-goal advantage into the break.
Andre Boucaud replaced Dan Sweeney at the interval but his time on the field was short-lived after picking up an injury and making way for Ephron Mason-Clark on 53 minutes.
Coulthirst nearly found the net again just past the hour mark after latching onto Cheye Alexander’s free-kick but his header whistled over.
Wes Fonguck nearly marked his England C call-up with a goal on 65 minutes, forcing Dudzinski into a strong save after bending an effort that looked to be floating into the top right corner.
Tyler Cordner picked up a yellow for the visitors with just over 20 minutes remaining before substitute Robinson equalised with a stunning bending effort from the edge of the area seconds later.
However, The Bees regained their lead soon after as Coulthirst bagged his brace after curling in a strike that nestled the bottom right corner.
Two dangerous chances in quick succession fell to Alexander and Mason-Clark respectively but Dudzinski did well to prevent the lead from extending.
The missed chances proved costly as Robinson was on hand again to ensure his side returned home with a point after a free-kick in stoppage time flew past Mark Cousins.
Still’s men travel to Wrexham on Saturday aiming to return to winning ways.
Bees Starting X1: Cousins, Alexander, Robson (C), Adams, Coulthirst, J.Taylor (Payne), Fonguck, Walker, Tutonda, Sweeney (Boucaud) (Mason Clark), Jules.
Unused Subs: Matrevics, Barham.
Goal Scorers: Coulthirst (P) 11’, 80’
Attendance: 965 (59 visitors)