Solar Elevation Angles -- Anomaly?
Your confusion may stem from the way we sometimes draw the Sun (not to 
scale) as though it's close to the Earth. 

In reality, as you note, the lines of vision to the Sun are nearly parallel: 

 

Note that the angles (measured from the red line, the horizon, to the 
arrow) are still different, but for entirely different reasons. 

I've actually stolen @Andy's answer (and @Tanenthor's comment), but 
I'm hoping the diagram makes it easier to understand. 

Re the parallel lines, the 11810x60 image below may help explain: even 
though the lines eventually converge (on an object about 10 times the 
distance to the moon, still nowhere near the distance to the Sun), 
they are nearly parallel when leaving Earth.