Open Source Job Scheduler with REST API

Question 1

JobScheduler would seem to meet your requirements:

Open Source see: Open Source and Commercial Licenses
Rest API see: Web Service Integration
Parallelism see: Organisation of Jobs and Job Chains

I think that these areas are also covered (I downloaded and trialled the application): See here

Tree like Job dependency
Hold & Release
Rerun failed steps

I'm not affiliated with SOS GmbH

Question 2

ProActive Scheduler is an open source job scheduler.

It is part of OW2 organization
It is written in Java so it comes with a Java and a REST API
It provides workflows that are set of tasks with dependencies and more (loop,replicate, branch), upon failures you can control if the task should be cancelled or restarted
Parallelism and distribution is at the heart of it, with features like for instance

Commercial Support is provided by Activeeon, the company behind ProActive (full disclosure: I work for Activeeon).

Question 3

You might be interested in DKron

Dkron is a system service that runs scheduled jobs at given intervals or times, just like the cron unix service but distributed in several machines in a cluster. If a machine fails (the leader), a follower will take over and keep running the scheduled jobs without human intervention. Dkron is Open Source and freely available.

http://dkron.io/

Question 4

While not open source, a more cost effective job scheduler solution with REST API and support for the features listed is ActiveBatch workload automation and job scheduling. I do work for the company (being up-front) but our customers love how they can easily extend their automated processes to connect to any application, any service, any server with our REST API adapter. You can get more information here: https://www.advsyscon.com/en-us/activebatch/rest-api-adapter