The parity problem is one of the best-known classification problems studied to examine the computational abilities of cellular automata. In this inverse problem, one is looking for a cellular automaton that can classify each initial configuration into one of two classes according to its parity. In the case of deterministic one-dimensional cellular automata, there exists a local rule that effectively solves the parity problem, but it is unknown whether it is the simplest possible rule. Specifically, it is known that a cellular automaton with a nine-cell neighborhood can solve the parity problem, whereas no cellular automaton with a five-cell neighborhood is capable of doing so. These findings have remained unimproved for the past 10 years. In this paper, we present novel tools that allow to narrow down the existing gap. With the help of these tools, we are able to demonstrate that there is no cellular automaton with a six-cell neighborhood capable of solving the parity problem.