I believe the monster is a giant mutated rat. The reasons why are:
1. J.J. Abrams calls the monster the Parasite - Rats are parasitic animals.
2. The picture of the chef in the kitchen - Rats are always infesting kitchens, and in Japan they even eat rats.
3. Two of the fake titles for the film were "Cheese" and "Chocolate" which Rats LOVE to eat.
4. New York is known for it's rats.
5. If Slusho is the reason why the "parasite" is mutated and it is in the ocean, rats are in the sewer line in NYC, which leads to the ocean, meaning that rats could have drank it and became mutated.
6. In an interview with Movieweb.com, JJ Abrams says "I don't give a RATS what people think about this movie, I know THAT it will kick ass. - is he hinting something here?
7. What about that Roar? Rats don't Roar, do they? Actually, Rats emit short, high frequency, ultrasonic, socially induced vocalization during rough and tumble play. - As it says on Wikipedia.org
8. In the trailer someone yells "its alive, its huge" - people don't expect to see a rat of this size.
This is just my opinion, and here is the translation for the back of the Chef card:
Thank you for checking us out! We are Introducing this week's tasty one dish. Make sure you eat it cold!
Skinless chicken breast - 2, cut in halves
Sōmen noodles - 10 oz.
Watercress - 1/2 cup cut into small strips
Turnip - 1/2 cup, thinly sliced
Shiitake Mushrooms - 1/2 cup
Chicken stock - 1/3 cup
Sake - 2 tbsp.
Sugar - 1/2 tsp.
Deep Sea Nectar - 1 pinch
In a small saucepan, stir together 1/3 cup water, chicken stock, sake, and sugar. Chill it until it becomes cold. Grill the chicken breast on both sides for about 8 minutes, and then chill. Boil the noodles for about 3 minutes, and then run under cold water until chilled. Mix the watercress, radish, and mushrooms into the sōmen. Slice the chicken thinly and arrange on top of the sōmen mix. Just before you serve, put the deep sea nectar in the sauce and pour over the noodles generously.
Go Go Delicious Chef!