I have 5 test suites which are independent of each other. 
I have to run it against the same environment. Most of my test suites consist of API calls.
The test cases inside the suites should run in sequence as they are dependent on each other.
Is there any way we can run all the test suites in parallel via the pybot command?
The simple solution is using Jekins:

You could install Jeknins with robotframework plugin.You can have
two job run in parallel by default without any slave node.
Or you have have multiple slave node, then using tag in robot and node label to distribute job.  

Just set the parameter in Jenkins job build section, such as:

pybot --include tag1 test.robot for job1
then set pybot --include tag2 test.robot for job2.

Then trigger upstream job. you will get them running in parallel.
But still you need to make sure the file you accesing is locked by one of the testing job.
Is there any way to run robot framework test suites in parallel?