This paper proposes a new approach for proof number (PN) search, named job-level PN (JL-PN) search, where each search tree node is evaluated or expanded by a heavy-weight job, which takes normally over tens of seconds. Such JL-PN search is well suited for parallel processing, since these jobs are allowed to be performed by remote processors independently. This paper applies JL-PN search to solving automatically several Connect6 positions including openings on desktop grids. For some of these openings, so far no human expert had been able to find a winning strategy. Our experiments also show that the speedups for solving the test positions are roughly linear, fluctuating from sublinear to superlinear. Hence, JL-PN search appears to be a quite promising approach to solving games.