Since no platform or version is mentioned, I assume Unix/Linux and 11gR2.
First of all, check Running ASMCMD in Noninteractive Mode. This explains that we can use ASMCMD in a script so whatever we can dream of, we can script. In this case in order to copy files we can use
asmcmd command options
A simple example is shown here Running ASMCMD commands in a script
Change this example slightly to get it to copy all files in a local directory, also using this Using the ASMCMD cp command as input:
ls | while read FILE
asmcmd cp $FILE +data/orcl/datafile/$FILE
I am not sure how smart it is to run multiple copies in parallel. It all depends on what your hardware can handle. If you want to run in parallel, run the copies in the background but be a little careful, when running too much in parallel, it will definately cause a slower throughput than running in a sequential way.
For windows it will be slightly different, the loop construct will be different.
I hope this helps.