  Recently there has been a debate concerning the universal properties of the
phase transition in the pair contact process with diffusion (PCPD) $2A\to 3A,
2A\to \emptyset$. Although some of the critical exponents seem to coincide with
those of the so-called parity-conserving universality class, it was suggested
that the PCPD might represent an independent class of phase transitions. This
point of view is motivated by the argument that the PCPD does not conserve
parity of the particle number. In the present work we pose the question what
happens if the parity conservation law is restored. To this end we consider the
the reaction-diffusion process $2A\to 4A, 2A\to \emptyset$. Surprisingly this
process displays the same type of critical behavior, leading to the conclusion
that the most important characteristics of the PCPD is the use of binary
reactions for spreading, regardless of whether parity is conserved or not.
