We introduce and explore a model of an ensemble of enzymes searching, in parallel, a circular DNA strand for a target site. The enzymes performing the search combine local scanning - conducted by a 1D motion along the strand - and random relocations on the strand - conducted via a confined motion in the medium containing the strand. Both the local scan mechanism and the relocation mechanism are considered general. The search durations are analysed, and their limiting probability distributions - for long DNA strands - are obtained in closed form. The results obtained (i) encompass the cases of single, parallel and massively parallel searches, taking place in the presence of either finite-mean or heavy-tailed relocation times, (ii) are applicable to a wide spectrum of local scan mechanisms including linear, Brownian, selfsimilar, and sub-diffusive motions, (iii) provide a quantitative theoretical justification for the necessity of the relocation mechanism, and (iv) facilitate the derivation of optimal relocation strategies.