I will skip the origin of carlism because I supouse you are awared.

Before the CW, the carlist party was Comunión Tradicionalista, it was a far right party, but not like the fascism of those years and it was somehow popular, usuallt earning seats in the Congress. They also had a militia, el Requeté.

After the failed coup d'état of July 17, and the start of the war, carlism supported unanimously the rebels.

In 1937, Franco unified Comunión Tradicionalista with Falange Española de las JONS for creating FET de las JONS, the one party. This measure was critized by purists from both formations, but they were silenced.

So, when Franco seized Spain, the carlists were one of the socalled "Fracoism Families". In the 60s, the heir of carlism claims was Carlos Hugo de Borbón y Parma, and he started seeing how carlists students and workers were creating their own platforms outside Francoists or even more, oppossing the regime. And he started supporting a change in the party, for creating a progresive one.

In 1968, Franco forced the exile of the Borbón y Parma family. Then, Carlos Hugo in 1970 create the party. They even joined the Democratic Junta of Spain, formed by the communist party. So in 1975, Carlos Hugo is "crowned" as claimant of the carlists claims.

But his brother, Sixto de Borbón y Parma, taked contact with the tradicionalist veterans with the goal of getting rid of Carlist Party. And Comunión Tradicionalista, with the support of far right terrorists as AAA or Guerrilleros de Cristo Rey, provoked the death 2 of Carlos Hugo supporters (with also many injured) in the main carlist annual event, the Actos de Montejurra, in honor of the carlists militamen killed in the Civil War.

In the first eleccions of the new democracy, neither carlist branch earned a seat, becoming really minority parties. Carlist Party was founder of Izquierda Unida (United Left), a socialist coalition that still exists, along PCE and other parties. But they left after misunderstandings with the communists.

Comunión Tradicionalista was unified with other minor carlist parties to create Comunión Tradicionalista Carlista, usually going in coalition with other far right parties as AES, but still without representation.

Btw, Carlos Hugo died in 2010 after being diagnosed with cancer, so the carlists claims are for his son, Carlos Javier de Borbón y Parma, or for his brother.