The conflict in "The Park" by James Matthews revolves around the tension between the narrator and the park ranger. The narrator feels a sense of injustice and resentment towards the park ranger for enforcing rules and regulations that restrict his freedom and enjoyment of the park. The conflict escalates when the narrator defies the park ranger's authority and refuses to comply with his instructions, leading to a confrontation between the two characters. Ultimately, the conflict highlights the clash between individual autonomy and institutional control, as well as the struggle for power and agency in a public space.