I'm fairly comfortable with arithmetic and high school math overall, but I'm having trouble being 100% sure that $2^{\large {1+2^3}}$ should be parsed as 2^(1+(2^3)) and not as 2^((1+2)^3), because I've heard that
One must evaluate the base of an exponent before it is raised to the exponent.
As I understand that is actually true and the only way to support the second alternative is to claim that the power is $1+2^3$ (instead of $2^3$ alone) and that its base is $1+2$.
Can anyone confirm that the correct interpretation is 2^(1+(2^3)) and that it yields $512$?
Yes, your work is indeed correct.
Note also: $2^{\large(1+2^3)} = 2^1\cdot 2^{\large(2^3)} = 2 \cdot 2^8=2\cdot 256 = 512$, just as an exercise in working with exponents.
This can be useful, e.g., if you are working with $3^{\large(2+2^2)} = 3^{2 + 4}= 3^2\cdot 3^4 = 9\cdot 81= 729.$
Which ever you find easiest to do in various such examples, if one has $a^{b+c}$, one can add $b+c = d$, then calculate $a^d$, or, equivalently, one can multiply $a^b\cdot a^c$.