The time period you are talking -- right after WWII -- was a boom time for divorce. While the return of veterans did lead to many more marriages, a Census report in 1949 encapsulating the decade noted that while 3 million more marriages happened than expected, 1 million more divorces happened as well.

There was indeed no such thing as no-fault divorce (although an organization called the National Association of Women Lawyers kicked off an effort in 1947). But the important thing was that laws were inconsistent across states. Only 24 allowed "absolute" divorces where marriage was severed but allowed remarriage. The main grounds at the time were (quoting The American Family: A Factual Background from 1948)

• violent or gross conduct
• menacing life, health, or happiness
• conduct showing willful or negligent disregard of marital obligations
• incapacity to fulfill marital obligations
• civil death through criminal status or prolonged absence
• defective marriage

with a few specialized expectations specific to the wife or husband, like the wife being pregnant by an undisclosed person, or the husband's desertion. Again, if all six or some combination was included varied state by state. Some states were liberal enough they actively competed for potential divorcees (Nevada, Idaho, and Arkansas) although this was overall only a small number of divorces.

Even more importantly, there was written law, and there was actual practice. Reginald Heber Smith (part of an ABA group studying the divorce issue) wrote in 1947 that "society, by and large, has practically made up its mind to ignore the law." While there were strict courts, it was not uncommon for courts to look the other way. In a situation were both parties were amiable to the separation, a divorce would not be hard to find (and even if some difficulty was raised, a little light perjury could fix the problem). Due, essentially, to popular demand, there was a shift from thinking of divorce in terms of a "punishment" model to a "treatment" model.

In summation: yes, people who wanted to get divorced after WWII without invoking any real fault could do so. If our fictional couple had trouble they could go to Nevada to ensure a liberal court (they only had a 6 weeks residency requirement, lowest in the nation).

...

Now, for the reaction to the movie itself --

Despite The Parent Trap (1961) being well-known for having remakes after (including one with Lindsey Lohan) it was actually a remake of a remake of a remake. The original was a book, Das doppelte Lottchen (1949; the girls were nine, the parents have rather less money, but the details are relatively the same). This was made into a film a year later (Das doppelte Lottchen) using a screenplay by the author, and the real-life life twins Jutta and Isa Gunther. It was well regarded and won Best Picture, Director, and Screenplay for the 1951 German Film Prize.

1951 also saw a Japanese remake (Hibari no komoriuta) and two years later came a British one (Twice Upon a Time).

In other words, the story had already had international appeal right from the start, and there didn't seem to be deep upset about the divorce. Meanwhile, in Hollywood, story of divorcees reunited generally met with approval, in movies like The Women (1939), Bedtime Story (1941), and Let's Make It Legal (1951).

So, mending pre-story divorces did not raise a fuss. What could raise a fuss was divorce on screen in the middle of a story -- From Here to Eternity famously had Karen (Deborah Kerr) consider having a divorce (with a famous beach scene between Kerr and Burt Lancaster) and the Catholic Church raised objections.

You know what probably happened? They must have quarrelled and parted, and just sort of bisected us — each taking one of us.

Returning to The Parent Trap, the context above meant there was nothing shocking about the divorce. It does, in a sense, take an anti-divorce stance, but in (what was then) a modern-for-1961 sense -- not pretending the problems were not there, but trying to address them: treatment rather than punishment. The twins Sharon and Susan (both played by Hayley Mills, who was a true star by then after her showing in Pollyanna the year before, and part of the reason the age got bumped up from 9 to 13) meet at a summer camp, realize they are long lost siblings, switch places, and bring the parents back after fending off the father's fiancée using the power of love and rock 'n' roll. If anything, the use of rock music was the more controversial part -- in the 1950s rock still meant doom to the youth, and it is certainly possible its use in The Parent Trap was part of its mainstreaming, including a credits song:

If their love's on the skids,

Treat your folks like kids,

So to make them dig,

First you gotta rig,

The parent trap.

...

Aquila, M. L. (2014). Movies as History: Scenes of America, 1930-1970. United States: McFarland, Incorporated, Publishers.

Brode, D. (2014). From Walt to Woodstock: How Disney Created the Counterculture. University of Texas Press.

Oren, L. (2018). No-fault divorce reform in the 1950s: The lost history of the “greatest project” of the National Association of Women Lawyers. Law and History Review, 36(4), 847-890.

Smith, I. R. (Ed.). (2017). Transnational film remakes. Edinburgh University Press. (Specifically, Chapter 8 by Constantine Verevis, and this is the best source if you're wanting lots more detail on the movie, including the fact that the title came from a contest with a one-hundred-dollar prize. Walt Disney had wanted the title His and Hers but another company had already picked it for an entirely different movie released in 1961.)

Japan did return to the subject matter; one of the more fascinating remakes is the anime series Watashi to Watashi: Futari no Lotte from 1991.