Usually, trainer make Exeggutor learn Confusion or Zen Headbutt for fast move slot and Solar Beam for charge move slot. Currently no Pokemon can reduce damage of both glass- and psychic moves.
While Chariard's moves, such as Wing Attack and Ember, are both effective against to Exeggutor, Exeggutor's moves, such as Seed Bomb or Solar Beam, grass-type charge moves, are not effective against to Charizard. The damage of those moves are reduced.
To beat Exeggutor, find or evolve Growlithe into the one having Fire Fang as a fast move, and Fire Blast or Flamethrower as a charge move. Those fire-type moves are super effective against to Exeggutor.
Other fire-type Pokemon option
For list of fire-type Pokemon, tap the name of the type below.
While the damage of grass-type charge moves Exeggutor use are reduced, Pidgeot's flying-type moves are super effective against Exeggutor.
Exeggutor's psychic-type moves, such as Confusion and Zenn Headbutt, are not very effective against Hypno. Although the damage of Hypno's psychic-type moves are reduced as well, its ghost-type charge move, Shadow Ball, is super effective against Exeggutor.
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I would think that Scyther would be the ultimate pokemon to take out Exeggutor. The combination of being a bug and flying pokemon, both resistant to grass types, while using bug and dark attacks, which are super effective against grass and psychic types respectively, means that you're giving increased damage while receiving very little. Other good pokemon would be a high level Butterfree (bug/flying) that only had bug attacks, Pinser with bug attacks and Parasect has already been mentioned. Do you see a theme here, go with the bug.
Parasect is fantastic for beating Exeggutor as long as it has two bug attacks. Bug is very effective against both psychic and grass (meaning double attack bonus), while bug/grass typing makes it resist psychic and double resist grass attacks. I have a maxed Parasect at about 1,350 and it can easily defeat Exeggutors over 2,200 with some good dodging.