The　simulation　of　unsteady　incompressible　fluid　flow　and　heat　transfer　problems　is　a　huge　time-consuming　task　and　has　been　becoming　one　of　the　hottest　topics　in　the　fields　of　computational　fluid　dynamics　and　numerical　heat　transfer.　Pure　spatial　parallelization　has　been　so　far　the　most　widely　used　parallel　strategy　to　speed　up　unsteady　simulations　but　suffers　from　the　problem　of　speedup　saturation　as　parallel　scale　increases.　In　order　to　further　speed　up　unsteady　simulations　and　exploit　the　concurrency　in　temporal　dimension,　we　propose　a　novel　temporal　parallelization　strategy,　i.e.,　parallel　inverted　dual　time　stepping　method,　by　simply　inverting　the　sequence　of　the　inner　loop　(pseudo　time)　and　the　outer　loop　(physical　time)　within　the　dual　time　stepping　framework.　The　parallel　performance　of　the　proposed　method　is　verified　and　validated　with　three　steady/unsteady　problems.　Parallel　results　of　the　proposed　method　agree　well　with　those　of　the　dual　time　stepping　method　and　the　available　benchmark　solutions.　The　parallel　efficiency　of　the　proposed　method　is　about　96%,　81%,　62%,　51%,　45%　for　4,　8,　12,　16,　20　CPU　cores,　respectively,　which　is　considerably　higher　than　that　of　the　widely　used　parareal　method.　The　proposed　method　has　potential　in　further　increasing　parallel　scale　as　spatial　parallelization　is　saturated.　©　2020　Elsevier　B.V.