Probability calculation involving 2 wheels and sum of numbers being even

There are 2 wheels, with numbered regions, each wheel has a pointer to select a number that is written on the numbered region. One wheel has 4 numbered regions consisting of 2 odd (say 3, 7) and 2 even numbers (say 2, 6) and the 2nd wheel has 3 numbered regions consisting of 2 odd (say 5, 9) and 1 even number (say 4). if you spin both the wheels once, what is the probability that the sum of the two selected numbers is even ?

As an initial answer, to get an even number as the sum the numbers selected from each wheel should have the same parity (both either even or odd).

First case is that both are even, for the first wheel the chance of an even number is 1/2 and for the 2nd wheel this is 1/3. Since these are independent turns is it correct to say that the probability of both even would be 1/6.

2nd case is that both are odd, for the first wheel the chance is 1/2 and for the 2nd wheel this would be 2/3. Again the probability of both numbers being odd would be 1/3.

So the net probability of these 2 events occurring would be 1/6 + 1/3 = 1/2. Is this correct ?