The hosts had the better of the chances in the first half but went in trailing after Kari Arnason struck just before the break.
The midfielder rose highest in the Coventry box to head home Damien Johnson's free-kick.
McIndoe wasted no time in drawing the home side level after the break when he coolly converted a Carl Baker through ball.
The hosts pushed on in search of a goal early on and Sammy Clingan had an effort from 25 yards deflected behind for a corner and from the resulting set-piece Clingan's delivery was missed at the back post by two attackers.
Plymouth had their first attack of any note after 10 minutes when Jamie Mackie picked the ball up midway in the hosts' half and after shrugging off a challenge drove the ball wide of the upright from the edge of the area.
Keiren Westwood was then forced to make a fantastic save from Mackie after he drilled a strong drive towards the top corner.
The hosts continued to have the better of possession but both sides failed to create any clear goalscoring opportunities.
Clinton Morrison saw his shot from inside the area go harmlessly wide before a weak Clingan shot was watched wide by Plymouth goalkeeper David Stockdale.
As the first half drew to a close Baker found himself free in the area but Plymouth's Yannick Bolaise made a last-ditch tackle to clear the ball behind for a corner.
The Pilgrims took the lead with the final attack of the first half when Arnason glanced a header into the net from Johnson's free-kick.
Sky Blues manager Chris Coleman made two changes at the break with Gary Deegan and Marcus Hall replacing Aron Gunnarsson and Stephen Wright.
The changes spurred the hosts into life and after just four minutes of the second half Baker chipped the ball through for McIndoe to race onto and he slotted low into the bottom corner to level the scores.
Coventry continued to push forward and Jon Stead almost gave them the lead with a turn and shot from the edge of the area but Stockdale got down to collect the effort.
Plymouth substitute Gary Sawyer almost made an immediate impact with a long-range effort but it drifted wide of Westwood's goal with half an hour left to play.
In the closing stages Pilgrims midfielder Carl Fletcher had an effort blocked by Hall and a Chris Clark cross almost caught Westwood out as Plymouth looked to find the winner.
Bolasie almost snatched the points in injury time but his volley was blocked en route to goal by Coventry's defence.