The　parallel　structure　in　the　reliability　engineering　is　widely　used　in　many　industrial　systems　such　as　power　generation　systems,　pump　systems,　production　systems　and　computing　systems.　Designing　the　optimal　maintenance　policies　for　these　parallel　systems　can　minimize　the　operation　cost　while　achieving　a　satisfying　system　availability.　Previous　studies　have　studied　various　maintenance　optimization　for　parallel　systems.　However,　little　research　has　considered　the　multiple　failure　modes　that　widely　exist　in　many　real　industrial　systems.　This　research　considers　a　parallel　system　that　consists　of　heterogeneous　components　subject　to　two　types　of　competing　failures:　a　one-stage　catastrophic　failure　and　a　two-stage　delay-time　failure.　Inspections　are　conducted　periodically　to　identify　the　state　of　each　component.　Three　maintenance　policies　are　proposed　in　this　research.　To　find　the　optimal　inspection　interval,　we　analyze　the　renewal　process　of　the　system　to　minimize　the　long-term　expected　cost　per　unit　time.　A　simulation　procedure　is　proposed　to　evaluate　the　long-term　expected　cost　per　unit　time.　Numerical　experiments　are　carried　out　to　illustrate　the　applications　of　the　proposed　maintenance　policies.　The　discussion　provides　useful　insights　on　choosing　the　best　policy　under　different　system　performance　requirements.