Well, to begin with, the equation of your parabola is wrong. The vertex is at (0, 3) and the focus is at (0, 0). The focus is below the vertex so the parabola opens downward. The equation of the parabola is .
Now, y'= -(1/6)x so that, at x= 1, the slope of the tangent line is -1/6. What is the equation of a line with slope -1/6 through (1, 35/12)?
For the normal line, the slope is the negative reciprocal of the tangent line. What is the equation of a line with slope 6 through (1, 35/12)?