Problem 432

5.D.

Let $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$ be a permutation of $1,2,3,4,5$ that meets the requirements.

First, for $a_{1}, a_{2}, a_{3}, a_{4}$, there cannot be two consecutive even numbers, otherwise, all numbers after these two would be even, which contradicts the given conditions.

Second, if $a_{i}(1 \leqslant i \leqslant 3)$ is even and $a_{i+1}$ is odd, then $a_{i+2}$ is also odd. This means that an even number must be followed by two or more odd numbers, unless the odd number following it is the last number.

Therefore, $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$ can only be even, odd, odd, even, odd. There are the following 5 scenarios that meet the conditions:
$$\begin{array}{l} 2,1,3,4,5 ; \quad 2,3,5,4,1 ; \quad 2,5,1,4,3 \text {; } \\ 4,3,1,2,5 ; \quad 4,5,3,2,1 . \\ \end{array}$$