Suarez battled valiantly and substitute Facundo Torres caused problems as Uruguay dominated as the game went on. But they could not get the all-important goal on a pitch that cut up quickly and made silky play difficult.
The draw extended Paraguay's unbeaten run to eight competitive games, their longest such streak since 2011.
The top four teams in the 10-team group qualify automatically, and the fifth-placed side go into an intercontinental playoff.
Marcelo Moreno scored twice for the home side, once five minutes into the match and then again with seven minutes remaining.