Calls and puts should have the same implied volatility. The implied volatility should describe that portion of the options price attributable to the movement in the stock, ie the implied volatility. If your implieds are different you have not done enough work to identify what is causing the imbalance.
The most obvious culprits causing calls and puts to have different IVs are interest and dividends. Interest rate assumptions can vary over stocks, expirations and even strikes. Stocks can be hard-to-borrow and instead of receiving interest for being short shares, interest is paid for the privilege of shorting these stocks. Since the hard to borrow-ness of a stock can change and usually fade over time, farther out expirations will have a lower hard to borrow rate than near months.
ORATS works hard to isolate the factors that cause an imbalance in call/put implied volatilities. First, we start with good dividend assumptions and we publish a popular dividend forecast file. Second, we solve for the residual yield that lines up the call and put implied volatilities. We even slope the residual yield as sometimes different strikes have different volatility assumptions.
Note the table below of Tilray (TLRY) expirations and especially the residual yield rate. This residual is the rate that lines up the call and put implied volatility and also shows a good picture of the implied hard to borrow rate for each month.
ORATS also offers an historical perspective of the residual yield rate or hard to borrow rate as seen here:
The other suspects causing an imbalance in IVs are liquidity related. Market makers will often have wider spreads on a high absolute delta option than the low one. For example, an in-the-money, low strike, high delta call, will likely have a wider spread than its partner put. Market makers will have to hedge the buying or selling of this call and that hedge usually involves buying or selling the underlying. The call will have more deltas to hedge and more risk to the market maker and this will often cause a wide bid-ask spread. Moreover, the market maker may determine that selling the deltas are harder than buying deltas.
All of the causes of IV imbalances can be isolated to give what should be got, call and put implied volatilities that line up.