What class of Devil Fruit does the Gomu Gomu no Mi come under? Considering the three types of Devil Fruits that has been explained are
1. Paramecia Type - where the user can get super human abilities and change their body parts into weapons, etc.
2. Zoan type - where the use can turn into a particular animal and inherit its abilities
3. Logia Type - where the user's body inherits a particular element and the body becomes like air which cannot be touched or damaged by non Haki users.

So what exactly is the type of Gomu Gomu no Mi?
It cannot be Zoan or Logia. It's very close to a Paramecia but still doesn't have a logical explanation to be of Paramecia type!
So is it some special type of Devil Fruit?
From the wiki.


  The Gomu Gomu no Mi is a Paramecia-type Devil Fruit that enables the user's body to stretch like rubber, making the user a Rubber Human.


You further ask..


  It's very close to a Paramecia but still doesn't have a logical explanation to be of Paramecia type!


Again from the wiki.


  Paramecia is one of the three Devil Fruit types. These fruits give the users a power that can affect their body, manipulate the environment, or produce substances. In general, Paramecia Devil Fruits are fruits that give their users powers other than transforming into elements, like Logias, or transforming into animals, like Zoans.