Consider the discrete logarithm problem over $\mathbb{Z}_p^{*}$ with generator $g$:
$$\text{Input: $y \in \mathbb{Z}_p^{*} \hspace{5em}$} \text{Output: $\log_g{y}$}$$ This problem is widely conjectured to be classically intractable for large $p$. Let's assume this is provably the case.
Now consider the following problem: $$\text{Input: $y \in \mathbb{Z}_p^{*} \hspace{5em}$} \text{Output: is $\log_g{y}$ even?}$$
The second problem is provably easier than the first because you can get $\log_g{y} \mod 2$ easily if you know $\log_g{y}$, but not vice-versa: $$\begin{align}(y,\;\log_g{y}) &\Rightarrow \log_g{y} \mod 2 \\(y,\;\log_g{y} \mod 2)&\nRightarrow \log_g{y} \end{align}$$
Despite this, can I make any assumptions about the difficulty of this second problem?
More generally, if I modify a classically intractable problem so that I require not the whole original output (e.g. $\log_g{y}$), but some property of it (e.g. the parity of $\log_g{y}$), are there cases in which the modified problem may remain classically intractable? How much can I speculate about whether this modified problem is classically intractable?