In the US, because it's almost impossible to pass legislation without support from both parties, legislation doesn't have a particularly high chance of being repealed, due to both parties having 'buy-in.'
In parliamentary countries, where one party wields exclusive legislative authority for a time, how do they deal with the possibility of their legislation being repealed? Has it historically been a problem? Is there generally a 'gentleman's arrangement' where parties don't repeal the other's work without strong cause?
In parliamentary countries, where one party wields exclusive legislative authority for a time
This is not true. In most parliamentary countries, the government is made up of a coalition of parties, none of which has a majority of their own. In Australia, even if a party (or coalition of parties, as is the current government) has a majority in the House of Representatives, they're unlikely to also have a majority in the Senate. This has happened only 4 times in the past 60 years. Most times, the government party has to negotiate with other parties in the Senate to get their legislation through. Therefore, very few governments in Australia have wielded exclusive legislative authority.
Is there generally a 'gentleman's arrangement' where parties don't repeal the other's work without strong cause?
Nope.
Going back to an example before the 20-year limit of this subreddit... The Australian Labor Party introduced universal healthcare in 1975, called Medibank. When the Liberal Party got into government later that year, they passed a law which privatised Medibank, turning it into a private health insurer in 1976. When the Labor Party got back into government in 1983, they restored universal healthcare under the new name of Medicare. There is no "gentleman's agreement" not to repeal a previous government's laws.