I had a dialogue. with someone. They sent something like:
When it comes to work, I like structure and being detail-oriented.
Should I call this a parenthetical phrase or preposition?
I thought hyphens should come only first what they modify, not after.
In your sentence, the first part is an example of an introductory clause.
When it comes to work is a dependant clause that introduces an independent clause. While the independent clause could stand on its own, the introductory clause gives it additional context.
(Even though I thought it was futile), I tried to lift the car on my own.
(Although it wasn't raining), I walked down the street with an open umbrella.
(Because I was depressed), I didn't leave the house all day.
In general, hyphens are used to prevent ambiguity.
She is a foreign-sales secretary.
Here, she is a secretary of foreign sales.
He is a foreign sales-secretary.
Meanwhile, he is a sales secretary who is also foreign.
In compound phrases that modify something, hyphens are generally only used when the compound phrase comes before what it modifies.
I had a half-hour session.
My session was a half hour.
In your example sentence, a hyphen would commonly not be used:
I am a detail-oriented person.
I am a person who is detail oriented.
