One can never be too clear on the logic of the comma. Common grammar advice seems partly responsible for the confusion. Many guides state a rule that commas should set off parenthetic elements and a separate rule that commas set off nonrestrictive clauses and phrases. Occasionally the guides equate nonrestrictive and parenthetical; commonly, they imply that parenthetic elements are a subset of nonrestrictive elements, but although parenthetic elements and nonrestrictive elements overlap, even seeming to coincide, the distinction expresses different bases for categorization.
Restrictive versus nonrestrictive distinguishes categorically, based on the way a modifier affects the meaning of the term modified, the modified term either restricted in scope or not, but parenthesis means "a remark or passage that departs from the theme of a discourse: digression." (Merriam-Webster's definition 1(b); I exclude 1(a) because it incorporates punctuation, the explanandum.) Parenthesis is a continuous rather than categorical concept, and it describes the pragmatics of usage rather than its semantics. A parenthetic element is parenthetic because it digresses from the writer's main line of thought.
Following these definitions, restrictive parenthetic elements and nonrestrictive nonparenthetic elements are possible. A parenthetic element that is restrictive digresses from the writers thinking yet changes the meaning of the modified phrase. A non-parenthetic non-restrictive element follows the writer's train of thought but changes the meaning of the modified phrase. Nonrestrictive/parenthetic and restrictive/nonparenthetic tend to correspond, but the correlation is only partial.
So what are examples of restrictive elements that are parenthetic?
Forgetful doctors who were uninsured because they neglected to mail their premiums suffered as much as doctors who intentionally stopped their coverage.Imagine this sentence appears in an article extolling the importance of obtaining insurance. The relative clause departs from the main line concerning objective importance by elaborating on a cause of failing to obtain insurance.
And nonrestrictive elements that aren't parenthetic?
In one case the doctor, who failed to procure medical-malpractice insurance, became indebted for three million dollars when he lost a lawsuit.The relative clause is more related to the main thought of the article on insurance's importance because without it the insurance theme is absent from the sentence.
Some readers will disagree with the classification; parenthesis as concept is vague because the distinction is partly subjective. The subjectivity of parenthesis combines with the concept's fuzzy boundaries to disqualify parenthesis as a basis for punctuation.