The authors consider the examination of how the triangularity conditions on the Gelfand-Levitan kernel (1951) affect the nature of the potentials. The original triangularisation property for the one-dimensional problem led to local potentials. The triangularity conditions used here (which is the one-dimensional analogue of the three-dimensional conditions used in a previous paper) lead to simple non-local potentials, which because of their form we call parity-dependent potentials. The inverse spectral theory problem is solved explicitly for several types of spectral measure functions. Such solutions give parity-dependent potentials with complete sets of eigenfunctions in terms of elementary functions. Using these examples, it is shown that a rich spectral theory exists with some aspects strikingly different from those of the spectral theory for local potentials.