Are problems with parallelism more common today? What issues can arise from parallelism?
I wrote about six potential issues with parallelism here:


https://sqlblog.org/2009/03/21/six-reasons-you-should-be-nervous-about-parallelism


That was three years ago, and I'm sure other equally dangerous bugs have come up since I wrote that. I haven't gone through and validated any that have been reported as fixed, but the workaround is almost always to use MAXDOP 1.

And there's another one that is still an issue in SQL Server 2012, and that hasn't been fixed in CU1:


https://stackoverflow.com/questions/9523213/possible-bug-with-range-option-of-window-aggregates-and-parallel-plans-in-sql-se
http://connect.microsoft.com/SQLServer/feedback/details/728932/parallelism-bug-with-sum-over-and-range


The workaround in that case is, again, to use MAXDOP 1.

There are other issues that can be blamed on parallelism that aren't really parallelism's fault. For example if your statistics are way off you can observe behavior where parallelism is employed but a single thread is doing all the work. This is the case where people sound the alarms because they're seeing CXPACKET waits and they think that parallelism is causing the problem, when in fact it is the victim. The knee-jerk reaction in those cases is typically to employ MAXDOP instead of investigating further and resolving the statistics issue or other root cause.

It's not clear what you mean by "too much parallelism" or whether your deadlocks and other issues are in fact related in any way to parallelism. In general I agree that parallelism is a positive thing but it is important to be aware of the potential issues above and also to be able to discern symptom from cause.