You are right, adding a comma after "that" does make it a parenthetical element (nonrestrictive clause). And a parenthetical element in the middle of a sentence is set off by a pair of commas (the exception is when the element is too short).
The sentence still makes sense ... but important information has been omitted ... I'm not very sure.
Yes, even if you omit the parenthetical element, the sentence still makes sense. You are thinking about this the wrong way.
A parenthetical element adds important information, but it is not essential to understanding the sentence.
A descriptive phrase that is restrictive—that is, essential to the meaning (and often the identity) of the noun it belongs to—should not be set off by commas. A nonrestrictive phrase, however, should be enclosed in commas (or, if at the end of a sentence, preceded by a comma). In the first example, the descriptive phrase (“with the guitar over her shoulder”) is essential information that identifies which woman is the mother. In the second example, the identity of the person who turned to the drummer (“My mother”) is clear; the fact that she has a guitar over her shoulder is not essential information.
- The woman with the guitar over her shoulder is my mother.
- My mother, with her guitar over her shoulder, turned to the drummer and gave the signal to begin.
Chicago Manual of Style (17th ed.): Commas with Relative Clauses, Appositives, and Descriptive Phrases
In your case, your main statement/argument/observation is this:
He thought that he wouldn't be able to fit in the chair. [This is stating what he thought]
And now you want to say why he thought what he thought.
Perhaps he thought that he wouldn't be able to fit in the chair, because his parka was too thick.
Perhaps he thought that, since his parka was too thick, he wouldn't be able to fit in the chair.