Parallelism, Use of auxiliary verb

To quote this article on Grammarly, parallelism in grammar is like parallelism in geometry: they face the same direction and never meet. So the clause "if wells can be dug there" and "[if] environmental concern addressed" are two sentences which both relate to "oil in Antarctica may be worth drilling for", but do not have much in common on their own.

As for the auxiliary verbs, here is a list of auxiliary verbs in English. Maybe you can clarify what the purpose of this exercise was, and where you had to find the auxiliary verb.

You say you do not understand "because "environmental" is a[n] adjective and "wells" is a noun"; you have to understand that parallelism bears upon clause structure and that the units that make up the clauses must show a certain uniformity (parallelism); the following are items of importance in this subject (agreement and parallelism).

• Verb Forms - Olympic athletes usually like practicing, competing, and to eat ice cream sandwiches. (wrong)

• Nouns vs. Verbs - For dinner we like lamb chops and to fry brussel sprouts. (wrong)

• Use of articles - Parallelism requires that an article (a, an, or the) or a preposition applying to all items in a list either appear before the first item only or be repeated before each item. Here is an example of the rule:

  Nonparallel: We can pay with a mark, a yen, buck, or pound.

• adverb and prepositional phrase - The detective deftly and with pizzazz outlined how the crime had been committed. (wrong)

However, this principle is not applied to the extent that you envisage: one of the noun phrases can comprise no adjective while another does so; the principle of parallelism would otherwisebe too restrictive.

• Athletes like long vacations, travel and frequent competitions.

• (wrong) Athletes like long vacations, travel and to compete frequently.

At current prices, oil in the Antarctic may be worth drilling for, if [wells can be dug there] and [environmental concerns ___ addressed].

The sentence contains what is known as 'gapped coordination', where the middle part of the second coordinate is missing but recoverable from the corresponding part of first coordinate.

The gap, marked '____' , is understood by reference to the first coordinate; in this case "can be".

The fact that gapping is appropriate here demonstrates that the coordinates have parallel structures.