Actress Molly Parker likes the moment when it all falls apart, when a character loses everything. “Because, do you just give up on life?” she said. “Do you just die? Or do you change?”

Parker (“Deadwood,” “House of Cards”), 52, had thyroid cancer about a decade ago. She doesn’t speak about it often, and she is quick to say that she is fine now. Dying was not really on the table. But the treatment did disturb her endocrine system. The recovery took years. And it changed her, she said.

Cerebral, with an often febrile intensity, Parker is the rare actress who can run both very hot and quite cool. She seems to feel more than the average woman and to think more, too. In her 30 years in the business, she has typically gravitated toward its margins — indie films, prestige television when prestige television was new. (That said, she has made one Hallmark movie — for the money, she stressed.)

Which is to say, she was not looking to star in a network show, let alone something as blindingly normal as a medical procedural. If the role had been offered to her 10 years ago, she wouldn’t have considered it.

“I was only going to do indies for the rest of my life,” she said.

But when her agents sent her the scripts for “Doc,” a Fox series that premiered Jan. 7, she read them. Yes, it was a network show. Yes, it was a procedural. Yes, it would mean time away from her teenage son. Yet she was drawn to the story of Amy Larsen, a brilliant doctor who gets retrograde amnesia after a serious car accident. In Amy, she found another woman at a moment of crisis, another woman who would choose to survive.

I met Parker on an afternoon in early December. She had traveled from Los Angeles, where she lives, to New York City for a couple of days of publicity events. But she had found an hour or two to stop by Canada gallery in Chinatown. Glamorous in a fur coat, her voice soft and surprisingly low, she wandered the few rooms, admiring some photo collages by Lee Mary Manning and abstract paintings by Lily Ludlow, a friend. She had helped to fund the space, opened by friends of hers, in its early days, using some of her first movie paychecks.

“ ‘Investor’ is a fancier word than what I was doing,” she said. “But I bought art from them, and that was how we started.”

Parker was raised in a small town outside Vancouver, British Columbia, and spent much of her childhood studying ballet. Although she was accepted into the biology program at a nearby college, instead she moved to Vancouver, waiting tables and taking acting classes. In 1996, she starred in the indie film “Kissed,” playing a necrophiliac mortuary student.

“It was the most fun I’ve ever had,” she said.

For a decade, she went from one indie film to the next, partly out of snobbery, partly because the big studios weren’t yet interested. “I was maybe a little too intense,” she said. But she liked the life, traveling all over the world making what she called “weird, weird, arty films.”

She was drawn to darkness, to characters in extremis — addicts, sex workers, women undergoing extreme loss. In her private life, she was quiet, introverted to a fault. At work she could be more expressive, and for a long time that felt right.

Although she appeared in a little-seen Canadian series, “Twitch City,” her first major TV role was a two-episode arc as a rabbi in “Six Feet Under.” A year or two later she auditioned for “Deadwood,” David Milch’s revisionist Western series for HBO. Beginning in 2004, she spent three seasons as Alma Garret, a laudanum addict in possession of her murdered husband’s gold claim.

Timothy Olyphant, who played her lawman love interest, recalled how easy it was to act opposite her. “Usually someone as good as she is, you expect there to be some drama,” he said. But there was no drama, only a seeming effortlessness. “She was both in complete control of the scene and at the same time completely open to it going somewhere unexpected,” he said.

After “Deadwood” was abruptly canceled, Parker tried on a network show (“Swingtown”), but it didn’t take, and she had roles in “Dexter” and “House of Cards.” In the years after her cancer treatment, her choices — which included a live-action “Peter Pan & Wendy” and the Netflix remake of “Lost in Space” — became somewhat less predictable.

That an actress of her indie proclivities would choose to head a network show still comes as a surprise. And if “Doc” has a few idiosyncrasies — the show is based on an Italian series, which is based on a real-life case of a clinician with amnesia — it is still very much in the mold of a standard medical procedural. (At one point a character tells Parker’s Amy: “You’ve always been a maverick unafraid to take chances.”)

Each episode runs on two tracks as Amy solves medical mysteries while also contending with the larger mystery of who she became in the years she can’t remember, which excited Parker.

“She gets to discover who she is,” she said of Amy. “She can be anything.” This corresponded to her belief that humans are naturally multifarious, or as Parker put it, “There’s no limits to how weird people are.”

“Doc” views Amy prismatically — as a mother, a lover, a co-worker, a friend, a boss, an ex-wife. Barbie Kligman, the showrunner of “Doc,” had worried about casting someone to play this. “We needed the impossible,” she said. “This actress had to play all those levels at the same time.” And she had to make Amy sympathetic even as Amy says and does terrible things; that prism includes some very hard edges. Kligman felt that Parker could play all of those facets and was delighted when she agreed.