Suggestion: Count the even and odd numbers separately.
Tiago writes down all four-digit numbers with non-zero distinct digits that have the same parity. What is the probability that, if we choose one of these numbers, it will be even?
The four digits chosen are part of the sets $A=\{1,3,5,7,9\}$ or $B=\{2,4,6,8\}$.

With the elements of set $A$, we have 5 possibilities for the first digit, 4 for the second, 3 for the third, and 2 for the fourth, totaling $5 \times 4 \times 3 \times 2=120$ numbers with 4 distinct digits.

With the elements of set $B$, we have 4 possibilities for the first digit, 3 for the second, 2 for the third, and 1 for the fourth, totaling $4 \times 3 \times 2 \times 1=24$ numbers with four distinct digits.

Thus, it is possible to form $120+24=144$ numbers. Of all the possibilities calculated, only those generated by set $B$ are even numbers.

Therefore, the probability required is 24/144 = 1/6.