To avoid having to duplicate a lot of custom pipeline configurations it would be great to be able to specify parameters that has to be provided when manually triggering a custom build.

A common example would be deployment to different environment where the pipeline script is identical for every environment except for a few variables that change.
As it stands I have to create multiple custom pipelines to manage these environments rather than just creating one and specify parameters that are passed to the pipeline.

It should be possible to specify parameter definitions on Team and Repository level, just like environment variables. You then specify in your pipeline which parameters can be set, which are required etc. Default values should be overridable in the pipeline configuration.

Support for at least text and select list parameters should be possible.

For reference, TeamCity has a pretty good way of handling parameters that might serve as inspiration.