The answer to this is dependent on the local political histories of each of the nations, and a detailed discussion of the circumstances of each Western country would run very long, so I’ll try and speak broadly about the prevalence of Parliamentary systems in Europe. A key contributing factor to these is monarchy - with the exception of France, most of Western Europe entered the 20th century with monarchies, albeit with varying levels of actual influence over policymaking. As such, legislatures had existed in many of the western countries for hundreds of years. In the case of the United Kingdom, the transition from absolute monarchy to modern parliamentary democracy was a gradual one, marked notably by the Civil War (1642-1651). As Parliament slowly gained power, the position of Prime Minister began to take shape, with Robert Walpole being generally regarded as the First Prime Minister (1721-1742). The position was not, and still is not, formally defined in legislation, but instead exists as a longstanding convention. In the centuries since Walpole, the balance of power has shifted effectively entirely to Parliament and away from the Monarch, but the Queen remains the formal head of State.

This British political history is also the main reason for the prevalence of parliamentary systems among the former Dominions (NZ, Australia, Canada) - the decolonisation process was generally a gradual transition of power to elected local bodies, who chose to keep the Queen as head of state following full independence.

Germany’s usage of a Parliamentary system is similarly reflective of their political history. The Weimar Republic (1919-1933) was set up as a Presidential System, but was widely viewed in the aftermath of WWII as being fundamentally flawed, and as having aided the rise of the Nazi regime. As such, the constitution writing process of the new German Republic was a careful process, with many powers that the Weimar President had being delegated to the legislatures.

France’s current system is usually described as being semi-presidential, with many characteristics in common with the American system but some notable differences (such as the cabinet being responsible to the legislative branch rather than the executive).

In general, the main reason for these notable differences is to do with monarchy; those states that maintained monarchies slowly transitioned power away from them to elected bodies, resulting in parliamentary systems. For those states created in the aftermath of revolution (France, United States), Presidential systems tended to be chosen.

Further discussion of the US case can be found here in an answer by /u/uncovered-history and here by /u/FugitiveDribbling