To assess both the parity and the maximum value of the given function $f(x) = \cos x - \cos 2x$, we proceed through the following steps:

Step 1: Determine the Parity of the Function

Given $f(x) = \cos x - \cos 2x$, we need to find $f(-x)$ to check its parity. We know that $\cos(-x) = \cos x$ and $\cos(-2x) = \cos 2x$ due to the even property of the cosine function. Thus, we have:
$$f(-x) = \cos(-x) - \cos(-2x) = \cos x - \cos 2x$$
This shows that $f(-x) = f(x)$, indicating that the function is even.

Step 2: Determine the Maximum Value of the Function

To find the maximum value, we first express $f(x)$ using the double angle formula and some algebraic manipulation:
$$f(x) = \cos x - \cos 2x = \cos x - (2\cos^2 x - 1)$$
$$f(x) = -2\cos^2 x + \cos x + 1$$
This can be rewritten as a completed square to make it easier to analyze:
$$f(x) = -2(\cos x - \frac{1}{4})^2 + \frac{9}{8}$$
In this form, it's clear that the maximum value of $f(x)$ occurs when the squared term is zero, which happens when $\cos x = \frac{1}{4}$. At this point, the value of $f(x)$ is:
$$f(x)_{\text{max}} = \frac{9}{8}$$

Conclusion

The function $f(x) = \cos x - \cos 2x$ is an even function, and its maximum value is $\frac{9}{8}$. Therefore, the correct answer is:
$$\boxed{D. \text{Even function, maximum value is} \frac{9}{8}}$$