Raid 10 doesn't have 'parity' it has mirrors.
RAID 10 is short-hand for RAID 1+0.
You will have, at minimum, 2 RAID 1 mirrors (groups). In each group there are 2 disks and the contents of the two disks within one group is exactly the same
Then you have a RAID 0 stripe across the two groups.
For more details -> http://www.thegeekstuff.com/2011/10/raid10-vs-raid01
You can lose one disk in both groups and survive but you cannot lose both disks in one of the groups. In short - it can give you *up to* 2 disks failure out of 4, but it can also die if you lose the wrong 2.
Raid 5 has enough parity data to cover the loss of a single disk and, while it has overhead, the parity calculations in Raid 5 are very fast/easy. When it comes to Raid 6 it's far more complicated with far more overhead using Galois Fields to compute parity but you can lose 2 disks no matter which two - but only 2.
RAID 6 is going to be more fault-tolerant than RAID 10 with a small number of disks because there are situations where losing 2 disks with RAID10 = data loss/array failure. There are no situations where losing 2 disks in RAID6 will destroy the array.
The issue with RAID 6 is that the larger the array gets - the more disks - the more likely you are to lose more than 2 disks. The parity data only protects you against 2 failures no matter how many disks you have. Your chance of disk failures goes up with more disks as well - the more you have, the more chances for failure.
It's a balancing act depending on what you need / want to do and how redundant you want it to be and how much capacity you can stand to lose in the process. RAID 6 is going to have a LOT more overhead so it will be slower to write than RAID10 but it should read pretty quick as parity isn't calculated on reads - just writes and rebuilds.