Difference between Appositives and Parenthetical elements

All appositive phrases are parenthetical elements but not all parenthetical elements are appositive. An appositive construction is a group of words (with no verb) which gives an emphasis to the immediately preceding word. On the other hand, a parenthetical element, non-essential information, can have many different forms among which the appositive phrase.

ie My dog, the best one in town, is an obedient dog.
The appositive phrase in red is the group of words which gives "extra" information on "my dog". It is also a parenthetical element because you can remove that phrase and the sentence would still make sense.

A parenthetical element can also be a non-restrictive clause as follows:

My dog, which I bought at the pet shop, is an obedient dog.
In red is a non-restrictive clause, non-essential information, and hence a parenthetical element. Note that the portion in red is a clause, not a phrase. This is because that red portion, as opposed to the one in the first example, has a verb.

As you can see, parenthetical elements have many different forms among which those above 2.