I hate writing intros so I'll just go straight to the team:
Skarmory@Leftovers
Trait: Sturdy
EVs: 252HP/232Def/24Spd
Impish Nature
Spikes
Roost
Brave Bird
Whirlwind
Role: The most important part of shuffling is residual hazard damage, which on my team will be set up by Skarmory. Skarmory usually can set up all three layers of spikes, then switch out to Dragonite so I can begin Parashuffling. Skarmory also has roost, whirlwind and brave bird as standard moves for physical walling later in the game.
Reason: I chose Skarmory over forretress because he has a reliable form if recovery and a better attacking move. Also he has taunt. As compared to ferrothorn and donphan, two hazards are useful if I want to build up residual damage while shuffling, with spikes and stealth rocks rather than just one alone.
Dragonite@Leftovers
Trait: Multiscale
EVs:252HP/4Def/252SDef
Careful Nature
Thunder Wave
Dragon Tail
Roost
Substitute
Role: Dragonite is obviously the main focus of this team because of his Parashuffling set. First he thunder waves the opposing Pokemon (assuming it not very threatening) then set up a sub or dragon tails right away, causing the incoming Pokemon to take hazards damage and then repeats the process.
Reason: I chose dragonite because he's probably the best parashuffler around.
Gyarados@Leftovers
Trait: Intimidate
EVs: 252HP/252Def/4SDef
Impish Nature
Rest
Sleep Talk
Waterfall
Roar
Role: Gyarados is my secondary shuffler. After dragonite is down Gyarados can still be effective at shuffling through the opponent's team and causing hazard damage. Rest is used as a reliable form of recovery besides leftovers, waterfall for a filler/STAB move, and sleep talk to continue shuffling (hopefully) while it's asleep. Gyarados uses roar over dragon tail now because of 100% accuracy, no STAB, and phasing out subs, where dragon tail does not.
Reason: First off, he was recommended to me, and also he works well at weakening the opponents team before I attempt a sweep with Heatran or Conkeldurr.
Jirachi@Leftovers
Trait: Serene Grace
EVs: 80HP/252Atk/176Spd
Jolly Nature
Thunder Wave
Iron Head
Fire Punch
Wish
Role: Jirachi's main purpose is to troll people with para-flinch. This thing has caused a ton of forfeits. Jirachi can thunder wave things that dragonite missed, then iron head them and para-flinch to oblivion. Wish is for healing, and fire punch is to attack unsuspecting steel types like scizor, ferrothorn, and forretress.
Reason: I chose Jirachi because for one, I like it when people forfeit against me, and it is very effective at slowly kill walls and sweepers alike.
Heatran@ Air Balloon
Trait: Flash Fire
EVs: 4HP/252SAtk/252Spd
Modest Nature
Fire Blast
Stealth Rock
Earth Power
Hidden Power [Ice]
Role: Heatran is my main special sweeper, even though he is rather slow without a timid nature, paralysis on my opponent's Pokemon makes up for it. The moves are fairly straight forward, fire blast is his main attacking move, Stealth Rock gives even more hazards damage, so Heatran may come in begone dragonite does. Earth power is for coverage (mostly against other Heatrans) and HP [Ice] covers dragons.
Reason: I chose Heatran because he is a great offensive Pokemon, and with the element of surprise as most people will expect a bulky one. I chose modest nature because by the time Dragonite's dead, almost the whole opponents team will be paralyzed so Heatran won't need the extra speed.
Conkeldurr@Leftovers
Trait: Guts
EVs/ 120HP/252Atk/136SDef
Adamant Nature
Bulk Up
Drain Punch
Mach Punch
Payback
Role: Conkeldurr takes advantage of hopeful para haxes, and sets up a couple bulk ups, then he can do a slow sweep with drain punch. Mach punch is for things that may not be paralyzed, or if I'm not feeling lucky and just want some extra damage on the opponent. And payback is for coverage.
Reason: I chose Conkeldurr because he is my favorite Pokemon, and just so you guys know I will not replace him.

Skarmory@Leftovers
Trait: Sturdy
EVs: 252HP/232Def/24Spd
Impish Nature
Spikes
Roost
Brave Bird
Whirlwind
Role: The most important part of shuffling is residual hazard damage, which on my team will be set up by Skarmory. Skarmory usually can set up all three layers of spikes, then switch out to Dragonite so I can begin Parashuffling. Skarmory also has roost, whirlwind and brave bird as standard moves for physical walling later in the game.
Reason: I chose Skarmory over forretress because he has a reliable form if recovery and a better attacking move. Also he has taunt. As compared to ferrothorn and donphan, two hazards are useful if I want to build up residual damage while shuffling, with spikes and stealth rocks rather than just one alone.

Dragonite@Leftovers
Trait: Multiscale
EVs:252HP/4Def/252SDef
Careful Nature
Thunder Wave
Dragon Tail
Roost
Substitute
Role: Dragonite is obviously the main focus of this team because of his Parashuffling set. First he thunder waves the opposing Pokemon (assuming it not very threatening) then set up a sub or dragon tails right away, causing the incoming Pokemon to take hazards damage and then repeats the process.
Reason: I chose dragonite because he's probably the best parashuffler around.

Gyarados@Leftovers
Trait: Intimidate
EVs: 252HP/252Def/4SDef
Impish Nature
Rest
Sleep Talk
Waterfall
Roar
Role: Gyarados is my secondary shuffler. After dragonite is down Gyarados can still be effective at shuffling through the opponent's team and causing hazard damage. Rest is used as a reliable form of recovery besides leftovers, waterfall for a filler/STAB move, and sleep talk to continue shuffling (hopefully) while it's asleep. Gyarados uses roar over dragon tail now because of 100% accuracy, no STAB, and phasing out subs, where dragon tail does not.
Reason: First off, he was recommended to me, and also he works well at weakening the opponents team before I attempt a sweep with Heatran or Conkeldurr.

Jirachi@Leftovers
Trait: Serene Grace
EVs: 80HP/252Atk/176Spd
Jolly Nature
Thunder Wave
Iron Head
Fire Punch
Wish
Role: Jirachi's main purpose is to troll people with para-flinch. This thing has caused a ton of forfeits. Jirachi can thunder wave things that dragonite missed, then iron head them and para-flinch to oblivion. Wish is for healing, and fire punch is to attack unsuspecting steel types like scizor, ferrothorn, and forretress.
Reason: I chose Jirachi because for one, I like it when people forfeit against me, and it is very effective at slowly kill walls and sweepers alike.

Heatran@ Air Balloon
Trait: Flash Fire
EVs: 4HP/252SAtk/252Spd
Modest Nature
Fire Blast
Stealth Rock
Earth Power
Hidden Power [Ice]
Role: Heatran is my main special sweeper, even though he is rather slow without a timid nature, paralysis on my opponent's Pokemon makes up for it. The moves are fairly straight forward, fire blast is his main attacking move, Stealth Rock gives even more hazards damage, so Heatran may come in begone dragonite does. Earth power is for coverage (mostly against other Heatrans) and HP [Ice] covers dragons.
Reason: I chose Heatran because he is a great offensive Pokemon, and with the element of surprise as most people will expect a bulky one. I chose modest nature because by the time Dragonite's dead, almost the whole opponents team will be paralyzed so Heatran won't need the extra speed.

Conkeldurr@Leftovers
Trait: Guts
EVs/ 120HP/252Atk/136SDef
Adamant Nature
Bulk Up
Drain Punch
Mach Punch
Payback
Role: Conkeldurr takes advantage of hopeful para haxes, and sets up a couple bulk ups, then he can do a slow sweep with drain punch. Mach punch is for things that may not be paralyzed, or if I'm not feeling lucky and just want some extra damage on the opponent. And payback is for coverage.
Reason: I chose Conkeldurr because he is my favorite Pokemon, and just so you guys know I will not replace him.