  Large-scale magnetic fields thread through the electrically conducting matter
of the interplanetary and interstellar medium, stellar interiors, and other
astrophysical plasmas, producing anisotropic flows with regions of
high-Reynolds-number turbulence. It is common to encounter turbulent flows
structured by a magnetic field with a strength approximately equal to the
root-mean-square magnetic fluctuations. In this work, direct numerical
simulations of anisotropic magnetohydrodynamic (MHD) turbulence influenced by
such a magnetic field are conducted for a series of cases that have identical
resolution, and increasing grid sizes up to $2048^3$. The result is a series of
closely comparable simulations at Reynolds numbers ranging from 1,400 up to
21,000. We investigate the influence of the Reynolds number from the Lagrangian
viewpoint by tracking fluid particles and calculating single-particle and
two-particle statistics. The influence of Alfv\'enic fluctuations and the
fundamental anisotropy on the MHD turbulence in these statistics is discussed.
Single-particle diffusion curves exhibit mildly superdiffusive behaviors that
differ in the direction aligned with the magnetic field and the direction
perpendicular to it. Competing alignment processes affect the dispersion of
particle pairs, in particular at the beginning of the inertial subrange of time
scales. Scalings for relative dispersion, which become clearer in the inertial
subrange for larger Reynolds number, can be observed that are steeper than
indicated by the Richardson prediction.
