Put-Call Parity $C - P = S - K*e^{-rt}$ provided the implied volatility of $C$ and $P$ are the same.
If the implied volatilities are different, there could be arbitrage taking opportunities exist. However, it doesn't mean there must be an arbitrage opportunity. If the implied volatilities being different doesn't result in arbitrage opportunity, then the different implied volatility can exist in the market.
E.g. the implied volatility of $C$ can be lower than $P$, because by arbitrage we will try to execute reversal $C - P - S$, which is Ask - bid - bid. Here the spread cost us more. Similarly, if the implied volatility of $C$ is higher than $P$, we can try to execute conversion $P - C + S$, which is Ask - bid + Ask.
As shown, when executing reversal or conversion, because of spread, the two boundaries are different. It gives an upper boundary and a lower boundary of the difference of implied volatility between the put and call to float within.
As shown in the trading grid, the implied volatilities are indeed different:
