Suppose that for two integral-valued arithmetic functions $f_i$ ($i=1,2$), the following values are known: $$ \lim_{n\to \infty} \frac{ \{n:f_i(n) \text{ is odd}\}\cap \{0,1,\ldots, n-1\} }{n}.$$ (In other words the density of odd $f_i(n)$'s are known.)

Is there any result which relates these densities to the following density: $$ \lim_{n\to \infty} \frac{ \{n: \sum_{i=0}^{n} f_1(i)f_2(n-i) \text{ is odd}\}\cap \{0,1,\ldots, n-1\} }{n}?$$

Note that the sum inside occurs when multiplying the generating functions for $f_1$ and $f_2$. Indeed, $$ (\sum_{n=0}^{\infty} f_1(n)q^n)(\sum_{n=0}^{\infty} f_2(n)q^n) = (\sum_{n=0}^{\infty} (\sum_{i=0}^{n} f_1(i)f_2(n-i))q^n).$$

This occurred to me while studying the parity of certain functions.