  Topological nodal-line semimetals are characterized by one-dimensional Dirac
nodal rings that are protected by the combined symmetry of inversion
$\mathcal{P}$ and time-reversal $\mathcal{T}$. The stability of these Dirac
rings is guaranteed by a quantized $\pm \pi$ Berry phase and their low-energy
physics is described by a one-parameter family of (2+1)-dimensional quantum
field theories exhibiting the parity anomaly. Here we study the Berry-phase
supported topological transport of $\mathcal{P}\mathcal{T}$ invariant
nodal-line semimetals. We find that small inversion breaking allows for an
electric-field induced anomalous transverse current, whose universal component
originates from the parity anomaly. Due to this Hall-like current, carriers at
opposite sides of the Dirac nodal ring flow to opposite surfaces when an
electric field is applied. To detect the topological currents, we propose a
dumbbell device, which uses surface states to filter charges based on their
momenta. Suggestions for experiments and device applications are discussed.
