How can I override the "part of app-pipeline" buildname?

Put a stage('name'){} block in each parallel entry. The name of the stage will appear in the executor status. So name your stages whatever you want to see in the status.

Note, the "part of ..." label will still show in the build queue, but executor status will show correctly.

parallel (    'newOne': { stage('new-feature'){ //all the things } },    'second': { stage('second branch'){ //all the things } },    'third': { stage('third branch'){ //all the things } },)  

The executor will show

jobname #nnn (new-feature) jobname #nnn (second branch)jobname #nnn (third branch)

EDIT: I ran a test pipeline that is simulating a multiconfig job with 3 axes: OS, JDK Fruit. Each branch of the config combinations runs in parallel and has a named branch. The executor status indicates each combination running:

Try using:

currentBuild.displayName = "My friendly name"

Use:

currentBuild.displayName="${JOB_NAME} ${BUILD_NUMBER} (${BRANCH})"

If it a declarative pipeline, you need to wrap it with a script{}:

script{    currentBuild.displayName="${JOB_NAME} ${BUILD_NUMBER} (${BRANCH})"}