This study aims to investigate the dynamic relationship between oil prices and stock markets in the G5+ countries using Parkinson's proximal realized volatility. We separate positive and negative semi-variance to compute asymmetric aggregate static spillovers according to the Diebold and Yilmaz (DY) approach. Moreover, we use a Quantile VAR to investigate the behavior of series in different quantiles corresponding to different market scenarios. Consistently with the literature concerns, we use a daily sample of market indices prices with the Brent oil price from June 1, 2017, to July 2, 2022. We found an asymmetric linkage between oil prices and the stock market, which has significant implications for portfolio hedging strategies. Specifically, our research indicates that the impact of the Russian-Ukrainian conflict on the energy crisis has been significantly higher than that of the COVID-19 pandemic, especially in the short term. However, we observe a higher persistence of negative spillovers for COVID-19 compared to those recorded during the Russia-Ukraine war. From a methodological viewpoint, this result enforces the choice of an asymmetric model to investigate the volatility transmission between financial market series. Finally, we found crude oil to emit volatility spillovers in quantiles above 80%. This result emphasizes the instability perceived in crude oil price when general market volatility increase. Quite the opposite, about one-third of oil price shocks, are linked to the national stock exchange uncertainty in low and middle quantiles, underlining the investors' dependency on this commodity. These results have important implications for policymakers and institutional authorities, who must consider the changing macroeconomic environment and reduce dependence on Russian energy.