Put-call parity is a concept that prices options (of the same strike) in relation to each other such that there is no arbitrage possibility. This is because you can replicate calls with puts and stock, and the other way round.
For example, how do you replicate the PnL of an at the money long put?
For each dollar the stock price falls, the put is worth $100 at expiration.
We can replicate this by shorting 100 shares of stock; for every dollar that the stock price falls, we’ll make $100 too. However, the put becomes worthless as the stock price rises, but the short stock will keep losing money. Therefore, we buy an ATM call. This strategy is called a “synthetic put”
We can see that in both cases, the put and the synthetic put will have the same payoff; $100 for every dollar the stock falls, or $0 if the stock rises from now until expiration.
How This Applies to the Wheel
If you short stock and buy a call, that’s a synthetic put. If you do the opposite (buy stock and short a call), that’s a synthetic short put.
By now you should realize that when you’ve been assigned on your puts and you sell OTM calls, it is essentially the same as selling an ITM put. The only practical difference between the 2 strategies is margin utilization and the fact that traders typically sell CSPs at a strike lower than the current stock price while selling covered calls above the current stock price.
You should only wheel stocks that you think will have low volatility compared to IV (since you’re short gamma), you think IV will fall (you’re short vega), and you’re bullish on (since you’re long delta).
You should also be aware that covered calls and CSPs have the same risks, which include the risk of missing out on gains when the stock shoots upward (in which case you’ll have your premium and a few tears), or if the stock makes a large move downwards (in which case you’ll be bag holding 100 shares of stock).