In the text, it mentions that put-call parity (c - p = s - PV(X)) implies that vega on a call and a put with the same strike are equal as both the stock price and PV(X) both have zero vega as neither of them are directly related to volatility. However, isn’t stock price kind of related to volatility? Higher volatility = higher uncertainty = lower stock prices? Or is that not applicable here due to risk-neutral pricing? Help disprove me as I’m clearly wrong. 
Also, I was watching Mark Meldrum’s lecture when he mentioned that delta on options with the same strike price, even if they all have different expiry dates, are equal. Is this true? Correct me if I’m wrong, but the formula for d1, which is part of delta, includes T, which would change with different expiry dates.