Yes, the complexities of the family tree and the rival claims to the throne during the War of the Roses are confusing at best and impenetrable at worst! The scene you actually want to look at is Act II scene 5, when Plantagenet goes to visit his uncle, Mortimer, in the Tower of London. It is Mortimer who gives Plantagenet and us the history behind his father's execution.
Mortimer explains that his own imprisonment in the Tower of London and his father's death come from the same root. Mortimer tells his nepheew that his family next in line to the throne after Richard II, but because of the deposing of Henry IV, Henry's line ascended to power. When Mortimer tried to press his cause as the rightful heir, he was quickly thrown into jail. In response to this, Plantagenet's father raised an army to back Mortimer's claim to the throne and ensure he gained the crown. However, unfortunately, he was captured and executed, which led to the suppression of any claims to the throne from the Mortimers. Plantagenet thus replies that his father's death was the result of nothing more than "bloody tyranny," whereas his uncle cautions him to recognise the dominance of the House of Lancaster now.
I have included a link below to the enotes study guide section of this play, and in addition there is a link to a family tree that might help you steer your way through this play. Good luck!
Thank you so much this is amazing! So glad you answered. It's still complex but at least now I can study the correct information!