Two concepts are in play here:
- Calculation of moment of inertia for a composite section.
- Calculation of elastic beam stresses.
First, the calculation of $I_{total}$ for the wide flange section.
Your equation for $I_{total}$ via the parallel axis theorem is correct, but the execution went awry.
$$I_{total} = \sum (I + A \cdot d^2)$$
Since the section is symmetric, the centroid is at mid-height and the $Ad^2$ term will be zero for the web component. Thus:
$$I_{total} = 2*(I_{flange} + A_{flange} \cdot {d_{flange}}^2) + I_{web}$$
Since we're calculating moment of inertia about a horizontal line, let b = the width and h = the height of each component. Let d = the vertical distance from the composite centroid to the centroid of the component. Note that this is a totally distinct choice and concept from the distance used to calculate maximum stress.
$$I_{flange} = \frac{1}{12} bh^3 = 1,562 \ mm^4$$
$$A_{flange} = bh = 750 \ mm^2$$
$$d_{flange} = 147.5 \ mm$$
$$I_{web} = \frac{1}{12} bh^3 = 1.016 \cdot 10^7 \ mm^4$$
$$I_{total} = 4.2797 \cdot 10^7 mm^4$$
In your numerical calculation of $I_{total}$ it looks as though you squared the A terms instead of the d terms. It also looks as though you measured d to the outer edge of the flange instead of to the centroid. Note also that to convert from $mm^4$ to $m^4$, divide by $1000^4$.
An easy way to check your results is to use one of the many online moment of inertia calculators. You can find ones for an array of common cross sections, including wide flange sections.
Now the calculation of maximum bending stress
When we're assuming that plane sections remain plane and that the section remains elastic (an Euler-Bernoulli beam), the equation for pure bending stress is as you noted.
$$\sigma = \frac{My}{I}$$
The derivation of this equation should be available in any introductory mechanics of materials textbook. Wikipedia also contains a discussion of Euler-Bernoulli bending theory.
In this equation, $y$ is the distance from the neutral axis (our centroid). At the neutral axis, bending stress is zero. The maximum bending stress must therefore occur at the cross section height that is farthest from the neutral axis - at the extreme fiber. For the beam you sketched, the extreme fiber is located at the outer edge of the flange, $150 \ mm$ from the centroid. Because we are so often interested in the maximum bending stress, this particular distance of $y$ is assigned its own variable name - typically, $c$.
In design of steel beams, we often talk about the "elastic section modulus" which simply lumps together the equations you've looked at here, giving engineers a quick equation to calculate the maximum bending stress in the elastic section.
$$Elastic \ Section \ Modulus, \ S = \frac{I}{c}$$
This gives us a quick way to calculate the yield Moment, $M_y$.
$$M_y = \sigma_y \cdot S$$
Where $\sigma_y$ is the yield stress of the material.
Some notes on real-world steel beams
As Leafk noted, wide flange beams are usually designed to optimize the use of material, which typically results in flanges being thicker than webs. And while it's beyond the scope of this question it's worth noting that there are several other beam design considerations beyond pure bending stress.
