Differentiate to find the tangent: 2x=4pdy/dx so the tangent dy/dx=x/2p.
At (x₀,y₀), dy/dx=x₀/2p. This is the slope of the tangent line.
The equation of the tangent is y-y₀=(x₀/2p)(x-x₀), so y=(x₀/2p)(x-x₀)+y₀.
Note that the expected equation as given in the question cannot be correct because it doesn’t pass through the tangent point (x₀,y₀).