Question 726659
Let's place your (x,y) coordinate axis at the apex of the arch. Then we have, to start with
(1) y = ax^2
To find a we substitute the point (135,-182.25) into (1) to get
(2) -182.82 = a*(135)^2 or
(3) a = -182.25/((135)^@) or
(4) a = -0.01
Then we have
(5) y = -0.01x^2
This gives the equation of the parabola (arch) as if it started at the apex or maximum value of the parabola. However the description of the arch has x starting at the base which is 135m to left of (5). Therefore we must shift (5) to the right by 135 and get
(6) y = -0.01(x-135)^2
It also states that y (the height of the arch) is 182.25m above the starting pylon. Therefore we must also shift (5) up by 182.25m, giving us
(7) y = -0.01(x-135)^2 + 182.25
Simplifying (7) we get
(8) y = -0.01(x^2 - 2*135x + 135^2) + 182.25 or
(9) y = -0.01*x^2 + 2*1.35x - 0.01*135^2 + 182.25 or
(10) y = -0.01*x^2 + 2.70x - 0.01*135^2 + 182.25 or
(11) y = -0.01*x^2 + 2.70x 
Comparing the coefficient of (11) to
(12) y = ax^2 + bx gives
(13) a = -0.01 and b = 2.70
Let's check the equation of (11).
Is ( 0 = -0.01*0^2 + 0)?
Is (0 = 0)? Yes
Is (182.25 = -0.01*135^2  + 2.70*135)?
Is (182.25 = -182.25 + 364.5)?
Is (182.25 = 182.25)? Yes
Is (0 = -0.01*270^2  + 2.70*270)?
Is (0 = -729 + 729)?
Is (0 = 0)? Yes
Answer: The equation of the parabolic arch is y = -0.01*x^2 +2.70x.
PS The 50m and 19 steel cables do not enter into the problem.