Transaction Log Backups Serial or Parallel?
Should I set up one job that uses ALL_DATABASES? or is it better to set up one job for each database and kick them all off in parallel?


I would suggest to setup one job that would backup the transaction logs (serially). This would also make sure backup does not heavily utilities the I/O because you are running backup for database one at a time.

What could be possible drawbacks with running in parallel


Suppose you have 50 databases and you schedule transaction log backup of all the databases and they all start running in parallel this is definitely going to utilize lot of I/O. And if disk on which it is backing up files happen to have other data files you would see slowness. I have seen backup becoming slow when a poor query requesting lot of I/O runs along with backup job.
Again suppose you have 50 database would it not be difficult to manage 50 jobs in SQL Server agent and what would be condition if you have 100-200 databases I would just not like it when you open the SQL Server agent and see lots of job, just keep it simple. I am sure the same case would be with you. 



  The downside of serial would be that each successive backup waits until the other completes. This could potentially increase the amount of time between backups (ie, greater than 15 minutes).


Transaction log backups are mostly small and if you have a busy database producing lot of log records you might need to change the backup frequency. Mostly I have seen transaction log backup completing fine when frequency is 15 mins. I don't think should be matter of concern for you.


  Plus my concern would be that a failure in one backup stops the others from happening, and I wouldn't want that to be the case


I would say just don't worry about it. Transaction log backups just cannot fail unless you made some mistake. The mistakes can be


The owner running the job is removed from AD
Someone changed the recovery model of database.
Insufficient disk space


Apart from above I have not seen any reason for transaction log backup to fail. Its very robust you can rely on it.