I've created a script that can install SQL Server Express 2019 and 2022 in Parallels on M1. You can download it here:
https://github.com/jimm98y/MSSQLEXPRESS-M1-Install. Just run the bat file and wait until it completes.
It turned out the problem is not in the SQL Server, but in their installation. The script just fixes the installation so that the service can be started. I've been able to successfully use the SQL Express for development after installing it this way.
So far the only drawback I know about is that the LocalDB feature is not working from the ARM64 version of the Visual Studio. It looks like the ARM64 VS is trying to load x64 DLLs from the SQL installation, which obviously won't work. Otherwise, ASP.NET apps can connect to the SQL Server instance and I am also able to use the SSMS to manage the databases.