Both the Earl of Gloucester and King Lear distrust their children and cast them out—and both die for that lack of faith. Their stories are parallels of each other.

William Shakespeare's King Lear is a tragedy about love, trust, and family. It opens on the king of Britain asking his three daughters how much they love him; he intends to give the largest part of his kingdom to the one who loves him the most. When his youngest and favorite daughter Cordelia speaks, she says: 

Good my lord, you have begot me, bred me, loved me. I return those duties back as are right fit: obey you, love you, and most honor you. Why have my sisters husbands if they say they love you all? Haply, when I shall wed, that lord whose hand must take my plight shall carry half my love with him, half my care and duty. Sure I shall never marry like my sisters, to love my father all.

He casts her out for not professing her love as effusively as did her sisters. She leaves and marries the king of France.

Meanwhile, Gloucester is convinced by his bastard son, Edmund, that his legitimate son, Edgar, is plotting to kill him. Edgar escapes and disguises himself as a madman.

In the meantime, both men who rejected their children find out that the people they trusted weren't honest. Lear's other daughters reject him and take his kingdom. Gloucester is turned over to the duke of Cornwall, Lear's son-in-law, by Edmund. When Lear places his trust in Regan, Goneril, and their husbands, he condemns himself and his kingdom. In the way, Gloucester created negative circumstances for himself by trusting Edmund. He's blinded by Regan and her husband. 

In the end, the two men's stories still parallel. Both are saved by the children they rejected. In Lear's case, Cordelia brings her husband, the king of France, to fight on his behalf. They find him going insane from what's happened and attempt to nurse him to better health. In Gloucester's case, Edgar saves him from Oswald, who attempts to kill him. Edgar also later kills Edmund.

In the end, both men die. Lear dies of grief at Cordelia's execution. Gloucester dies from shock and happiness when he finds out that Edgar is still alive. 

The following is excerpted from the eNotes charcter analysis portion of "King Lear" which explains the inter-connection between the sub and main plot.

Gloucheter's situation parallels the situation of Lear. He mistakenly believes Edmund when the latter pretends to read a letter that is falsely said to be written by Edgar. In that letter, Edgar supposedly tells Edmund of his impatience to inherit Gloucester's estate. Gloucester, like Lear, responds emotionally, immediately denouncing his legitimate son (Edgar) and trusting in the son who really intends to do him wrong (Edmund). And like Lear, Gloucester is to be punished for his lack of insight or moral vision. That punishment comes in the form of a brutal incident wherein his eyes are ruthlessly plucked out by Cornwall. The physical blinding of Gloucester is symbolic of both his own and Lear's blindness to the truth about their children.

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