Here's an example from today that demonstrates my previous explanation about the relationship of put and call premium to a dividend.
CE closed at $113.98 today. The August $110 call closed at $4.40 x $4.90 and the Aug $110 put closed at $0.90 x $1.10 . The spreads are pretty wide and fair value is somewhere in the middle. I'm going to take the liberty of assuming that it's $4.58 for the call and $1.00 for the put. CE goes ex-dividend in two days for $0.54
The covered call writer collects the dividend and his potential profit is $1.14 ($110 + $4.58 -$113.98 +$0.54)
The writer of the short put receives $1.00 and assuming that he's receiving fair market interest for his cash (he didn't buy the stock), he receives about 14 cents of interest for a grand total of $1.14 which is the same as what the covered call writer received.
The uninformed put seller unaware of the pending dividend might look at this comparison (without including the dividend) and conclude that the put is fat and overvalued when indeed, it is not.
A reasonable objection to my option premium assumption might be that I'm simply making up what the fair value of the options is and I'm fabricating a correct answer based on that. In reality, option prices are linked by a formula for buying the stock called a Conversion (a Reverse Conversion or Reversal involves shorting the stock).
The Conversion involves buying the stock, selling a call, buying a put, and receiving the dividend (where both options are of the same series). The formula is:
+Call - Stock + Strike Price - Put + Dividend - Carry Cost = 0
If the options are fairly priced, the Conversion's profit is zero. Using all of the information from the above comparison of the covered call and the short put, including the assumed prices, it is indeed so.
This arbitrage also serves to provide liquidity to the market. If there is someone trying to sell the Aug $110 put another person trying to buy the Aug $110 call, if a floor trader or market maker can do a Conversion for a profit, he will take the opposite side of each of these two traders and offset his risk with a combination with the stock. Other than Pin Risk, it's riskless.
For a really over simplified explanation, if we pretend that the stock is at the strike price and there is no carry cost, then the formula simplifies to:
Call + Dividend = Put
IOW, the put's premium will always be greater than the call premium by the amount of the dividend.