From put-call parity: long call plus short put equals a forward. Long stock plus long put equals a call. Short stock plus long call equals a put.
Why desks care. Synthetics let you take or unwind exposure through whichever leg is cheapest or most liquid, without touching the underlying - useful when borrow is expensive, when a market is illiquid, or when a position needs to be adjusted without disturbing the primary market.
They are also how you check a quoted set of prices for internal consistency.