  In this work, we study the following basic question: "How much parallelism
does a distributed task permit?" Our definition of parallelism (or symmetry)
here is not in terms of speed, but in terms of identical roles that processes
have at the same time in the execution. We initiate this study in population
protocols, a very simple model that not only allows for a straightforward
definition of what a role is, but also encloses the challenge of isolating the
properties that are due to the protocol from those that are due to the
adversary scheduler, who controls the interactions between the processes. We
(i) give a partial characterization of the set of predicates on input
assignments that can be stably computed with maximum symmetry, i.e.,
$\Theta(N_{min})$, where $N_{min}$ is the minimum multiplicity of a state in
the initial configuration, and (ii) we turn our attention to the remaining
predicates and prove a strong impossibility result for the parity predicate:
the inherent symmetry of any protocol that stably computes it is upper bounded
by a constant that depends on the size of the protocol.
