A very nice solution, I like it a lot.
(Well... except that (30 x 5)+(40 x 5)+(50 x 5)+(60 x 5)+(70 x 5)+(25 x 5) is 1375 not 1325.)
My solution is different but you did more, you found the sum. My solutions shows that the sum is odd but your solution finds the sum and then you observe that it is odd. Here is my solution.
Just as you observed there are 5 odd numbers in the 30s; 31, 33, 35,37 and 39. Likewise there are 5 odd numbers in the 40s, 50s, 60s, and 70s. Thus you are adding 25 odd numbers. But the sum of an odd number of odd numbers is odd so the sum of the 25 odd numbers between 30 and 80 is odd.