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The Blues had taken the lead through an Aaron Burns penalty but had Glendinning dismissed for a foul on Declan O'Brien.
Sub keeper Mark Haughey saved Kevin Braniff's spot-kick but Glenavon did level in the second half when another penalty was scored by Andy McGrory.
At Shamrock Park, Portadown drew 0-0 with Carrick Rangers.
The match at Windsor Park was filled with drama and controversy.
Glenavon felt the penalty award against Conor Dillon in the 38th minute was harsh as the ball hit him at pace.
But referee Crangle pointed to the spot and Burns stepped up to make it 1-1.
The second penalty came on the hour mark, keeper Glendining seeing red for tripping O'Brien in the box.
Central defender Haughey, a former Glenavon player, went into goals and saved Braniff's poorly-struck penalty with his legs.
Glenavon got a second penalty on 80 minutes when ref Crangle ruled that Joel Cooper's volley had been handled in the box and this time McGrory drilled a strong shot past Haughey.
Linfield striker Andrew Waterworth had a great chance to score a winner but put a close-range header over from a Burns cross.
With Crusaders being held to a 1-1 draw by Warrenpoint Town, second-placed Linfield could have moved to within five points of the leaders.
As it is, David Healy's men remain seven points behind the champions, while Glenavon move up to third on goal difference.
Glenavon player-manager Gary Hamilton: "We are devastated that we are not going away with three points.
"I think we would have deserved it for our second-half performance. We had 70% of the play."
Linfield manager David Healy: "I thought their second penalty was harsh, so I am disappointed.
"Our players showed a lot of character in the second half and we deserved a draw at least"
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The Ports had Tim Mouncey sent-off before half-time in the match at Shamrock Park.
Midfielder Mouncey, who had been booked just four minutes earlier, got a second yellow card for a foul on Kyle Cherry.
In the second half Portadown were awarded a penalty when striker Marcio Soares was fouled by Aaron Smyth.
However, Mark McAllister's spot kick was well saved by diving Rangers keeper Brian Neeson.
Linfield had keeper Ross Glendinning sent-off as they drew with Glenavon and missed a chance to close the gap on Premiership leaders Crusaders.