To what extent does parliamentary system correlate to a decrease in
corruption?
There is some correlation, but this shouldn't necessarily be viewed as causation. For the most part, other factors predominate.
Parliamentary systems are basically the default choice. A majority of countries at all ends of the corruption scale are parliamentary systems (light green, orange, and red in the map below), so it is hardly surprising that countries at the top of the transparency scale are often parliamentary.
(Source)
The alternatives are basically Presidential systems (in blue and yellow in the chart above) and non-democratic or only partially democratic systems (purple, brown, dark green, and gray in the chart above).
Non-democratic systems
Non-democratic systems of government are, not very surprisingly, more prone to corruption. In part, this is because legitimate interests that would be accommodated through the democratic process can be ignored in a non-democratic system.
Why does Latin America have Presidential systems?
In the case of Presidential systems, one important factor is that most of the political systems in Latin America, which mostly have their roots in 19th century independence movements, chose to be Presidential systems rather than parliamentary systems, in imitation of the United States and their fellow Latin American early adopters of that system, rather than parliamentary systems. This was, in part, because the United States was one of the oldest democracies in the world and had a track record of success in the Americas.
Latin American countries, for a variety of reasons only dimly, if at all, related to the overall structure of their government, have had corruption in recent years. In part, this is due to being at an earlier stage of economic development. In part, this is due to the power of drug cartels in the illegal drug trade sourced in Latin America for a U.S. and global market.
The Philippines, which was previously a U.S. dependency, and Liberia was founded by freed American slaves, likewise owe their strong Presidential systems to their historical connection to, and modeling on, the United States.
Recent adoptions of Presidential or Semi-Presidential systems
A final consideration may have caused countries more at risk for corruption to adopt strong President systems, rather than Presidential systems causing corruption.
One of the reasons to have a strong President system is the fear that a parliamentary system will be incapable of generating stable coalitions to run the country that is indicative of a society that is ethnically or politically fragmented, rather than homogeneous.
For example, this was an important reason that Afghanistan's constitution after the U.S. deposed the Taliban (for twenty years) chose a Presidential system. It had deep political divisions and its constitution drafters were not confident that a majority could be consistently mustered to form a majority coalition.
This was probably also a factor in the many African Presidential and Semi-Presidential systems, and in Indonesia.
This fear is not unfounded. Countries with both parliamentary systems and deep political divisions, like Italy and Israel, have been famous for having highly fragmented and unstable coalition governments using that approach. A Presidential system, in contrast, affords stability in the executive branch, even in the absence of stable coalitions.
But a divided society without a culture of consensus regarding core political issues is also going to be one prone to corruption arising from those divisions.
Other factors
The top ten list cited in the question tends to reaffirm the intuition that countries low in corruption tend to have not just a parliamentary system but also to be small countries homogeneous enough to have stable political majorities and are "nation-states" with a well established national identity.
This certainly describes seven of the top ten countries: Denmark, Finland (Finnish identity trumps its ethnic divide between Finnish and Swedish speakers), New Zealand (there are Maori and Polynesian minorities but small enough to make coalitions feasible), Norway, Sweden, the Netherlands, and Ireland.
Notably, when Ireland was subject to non-democratic government by the English, and when the Netherlands was deeply divided between Protestants and Catholics on religious grounds, each of these countries had very high levels of corruption that faded in the case of Ireland when it secured independence and democratic government, and in the case of the Netherlands when its cultural divisions were worked out.
Size
All of the top ten nations except Germany are also small.
Smaller governments are arguably easier to govern without corruption than bigger ones, where tight control from the top by officials interested in stamping out corruption isn't as feasible.
Switzerland's canton system is also designed to decentralize decision making to make more decisions made by smaller and less diverse political units than the nation as a whole.
The federal system in Germany, similarly, decentralizes administration where corruption can arise as well to some extent. In contrast, Italy and France, which are similarly large countries that have historically had more centralized administrations have also had more corruption.
Political culture
Corruption is also often attributed to political culture, and it is also notable that all of these seven, plus Switzerland and Germany, have populations with Northern or Central European roots from the political bloc of "Western Europe" and are economically very developed. Switzerland is notably the oldest democratic system in continental Europe which has given it time to develop a democratic political culture.
Countries that are not economically developed and have only young democratic systems may be more prone to corruption and not have developed strong norms against it.
Shared cultural roots and homogeneity also can help reduce corruption by reducing the likelihood that there are divisive forces in government on fundamental issues that corruption might circumvent, even if there are political divisions on lesser issues.
This also reduces distrust of the government. Distrust of government can undermine its legitimacy and lead to corruption.
The Case Of Singapore
Singapore is the odd man out of the top ten, of course. It is ethnically overwhelmingly Asian. It is ethnically diverse and divided. And, while its current dominant party regime is arguably now democratic, its recent history is as a non-democratic regime. Comparisons can easily be made between Singapore, where even minor infractions are punished tyrannically, and Mussolini's Italy, in which the perennially corrupt country's fascist leader "got the trains to run on time."
This said, it is notable that rather than having merely a generic parliamentary system, Singapore has a political and legal system that are in the British tradition, rather than one rooted in civil law legal systems of continental Europe. Most countries, upon independence, adopted civil law legal systems, in part, because they don't require a large corps of professional lawyers trained for many years to discern the law from a large collection of case law precedents. But Singapore, in adopting a common law legal system also secured for itself a corps of senior civil servants socialized into the British legal and political tradition, and that "soft" human factor may have been influential in its ability to operate less corruptly.
