The experimental and numerical study of parallel one-dimensional arrays consisting of double-layer stacked small Josephson tunnel junctions is reported. In addition to several features already known for classic single-layer arrays, we find a new resonant state which corresponds to the locking of the fluxon oscillations to a frequency of small-amplitude oscillations of the superconducting phase in middle islands of small stacks. Numerical simulations of a simplified circuit agree well with experiments and confirm the suggested qualitative mechanism. In these experiments we find no indication for the so-called multiple-fluxon propagation in double-junction arrays.©1997 American Institute of Physics.