King Agnarr

The King and Queen of Arendelle are minor characters in Frozen. They are the parents of Elsa and Anna and were the rulers of Arendelle until their deaths.

Background
The King and Queen were the rulers of Arendelle, greatly loved and respected by their subjects and family. Together, they had two fair daughters, Anna and Elsa, and although Anna was a normal princess, Elsa was mysteriously born with powers of ice and snow. Apparently, the King and Queen were just as powerless as Anna, but it could be assumed magical abilities run in the family and merely skipped their generation or that something happened to the queen when she was pregnant with Elsa.

Judging by how he presented himself, the king was a mighty one, wise and kind, powerful and deligent. The queen appeared to be far more passive than her husband, though given the obvious motherly nature needed to raise two daughters. And with that, it was clear the rulers' primary priority were their daughters, willing to make large sacrifices to ensure their safety due to Elsa's unique abilities. And despite Elsa's powers, they didn't fear their daughter in the least, giving her just as much affection as Anna, and were even willing to make physical contact with her, but eventually decided against doing so, as Elsa preferred it that way, fearful she'd hurt them. And although they had faith in Elsa that she'll one day completely control her abilities, the fact that her life was as miserable as it was because of them devastated the king and queen, nearly driving the queen to tears at one point.

The Cursed Beginnings
The couple first appear after being frantically called by Elsa who accidentally struck Anna with ice magic while playing. They are worried about Elsa's powers getting too strong, but fortunately they find a map in the royal library that will take them to some trolls who can help Anna.

Anna, Elsa, the king, and queen head over to the valley, and the king of the trolls, Pabbie, informs them Anna can be healed, though they're lucky she wasn't hit in the heart, for it can be fatal. Pabbie then informs the royal family that, while Elsa's magic is beautiful, she should learn to control it and not let fear consume her, or horrible things will occur. The king assures they'll help Elsa learn to conceal her powers until they can help her gain control of them, and before they depart, Pabbie wipes Anna's memories of Elsa's abilities, believing it's best if she completely forgot of the mystical moments they shared, though rearranging the memories to make it seem as if they were playing out in the regular snow, as opposed to snow conjured up in the castle.

The two sisters grow up separately to further safety upon Anna as well as keeping the castle gates closed for the kingdom's safety as well and vice-versa for Elsa. As Elsa grows, her parents try to help her control her magic ice powers. The king tells Elsa, "Conceal, don't feel." as a way to control her powers and to wear gloves to avoid conjuring the magic. It doesn't work completely, because Elsa can still freeze things without trying.

Tragedy Strikes
One day, the King and Queen must set off on a two-week sea voyage. During a storm, they are killed at sea by monstrous waves. Anna and all the people of Arendelle mourn their deaths (a portrait of the couple is draped over with a black sheet of fabric), but Elsa doesn't attend the funeral out of fear of her powers being revealed and someone getting hurt as a result, even though as the eldest sister, she is the heir to the throne. With their deaths, Elsa's training of her powers was left incomplete, as her powers were still mostly out of her control.

The King still appears later in the film; in a painting of his own coronation seen hanging in the library of the castle when Elsa was trying to control her powers, and again when the evil Prince Hans betrays Anna in order to take over the kingdom, though the queen makes no further appearances.

Trivia

 * Many fans have noted that Anna and Elsa's father bears a striking resemblance to Walt Disney in his younger years, and many think this was done to lovingly allude to him. He also sounds a lot like Walt too, especially when he tells Elsa that she will be alright when he and his queen depart on their sea voyage.
 * The possibility of this being intentional is quite high, as many fans also notice parallels between Elsa's and Anna's relationship to that of Mickey's and Oswald's ( characters who are often referred to as Walt's children ).
 * Both were separated from each other when they were fairly young. ( Elsa having to be put in another room to keep her away from Anna. The rights to Oswald's character being given to Universal Studios )
 * Both had little involvement with each others' lives despite being related.
 * Both were reunited by not the best of circumstances, which was unintentionally caused by the main character. ( Anna accidentally unveiling Elsa's powers, Mickey learning of Oswald in Epic Mickey after accidentally unleashing the Shadow Blot on the Cartoon Wasteland )
 * Since their reunion the two have become close, vowing to always be there for each other. ( Elsa returning to Arendelle with Anna, Oswald fighting along side Mickey in Epic Mickey 2 )
 * It should also be noted that like the King, Walt had two daughters, Diane Marie Disney and Sharon Mae Disney.
 * The king also somewhat resembles Haakon VII.
 * In pictures, the king has dark hair and the queen has light hair, but in the movie, it's vice-versa.
 * The Queen's character model is a recoloured and customized version of Elsa's coronation model. There have been several differences, including her clothing and dark brown hair, but her face and eyes mirror her oldest daughter's.
 * The King and Queen are the second set of Disney parents to die around about the same time in the movie, the first being Tarzan's parents. Incidentally, Tarzan and his parents were also involved in a shipwreck, though they survived it.
 * Interestingly enough, both Frozen and Tarzan were directed by Chris Buck.
 * While healing Anna, Pabbie mentions how if Elsa allowed fear to consume her, horrible things would happen. The King then attempts to teach her to control her powers, but Elsa begins to live her entire life in fear of her powers, which only made them harder to control.
 * However, it appears this can be contradicted, as the statement that the King had told the young Elsa was "Conceal it, don't feel it, don't let it show", in order to keep her powers in check. But when Elsa grew older, she slightly changed it to "Conceal, don't feel, don't let them know." Hence, it is likely that she misunderstood her father's meaning over the years. Thus, it suggests that while the King was trying to teach Elsa to ignore her powers by acting like they weren't there so that she would not feel so isolated, Elsa believed she had to give up emotions to prevent her powers from being unleashed. As the years went by, Elsa's powers became harder to control due to the lack of experience from confronting her own emotions.
 * Some suspect that the Queen of Arendelle may be related in some way to Rapunzel's parents, due to certain similarities in appearance between the Queen of Arendelle and Rapunzel's mother, the Queen of Corona.