Solving/Optimizing a linear system in a finite field (Z/2Z)
(Note that a random assignment has probability at least 1/(1+$\hspace{.03 in}$(size($\hspace{.03 in}$Y$\hspace{.03 in}$)))

of making at least 1/2 of Y's entries equal 1. ​ That can be derandomized

by sequentially setting each variable to [something that forces at least

as many of Y's entries to 1 as it forces to 0, with ties broken arbitrarily].)

By this paper's proof of Theorem 5.6, for all real

numbers $\epsilon$, if ​ $0 &lt; \epsilon$ ​ then the promise problem

Input: ​ instance of your problem in which each row of A has exactly 4 ones

must output YES if: ​ ​ ​ there is an assignment which makes more than 1-$\hspace{.03 in}\epsilon$ of Y's entries equal 1

must output NO if: ​ ​ ​ ​ for every assignment, less than ​ (1/2)$\hspace{.03 in}$+$\hspace{.03 in}\epsilon$ ​ of Y's entries equal 1

is NP-hard.

When parameterized by the number of zeros in Y, your problem is obviously in W$[\hspace{-0.02 in}$P$]$O,

but I don't know anything else about your problem's parameterized complexity. 

This paper gives a possible try for your problem,

although it looks like it's just a student's "3rd Year Project Report".