If I have a SSL private key that is encrypted with a DES password, and I have to share with another admin a NODES copy of the private key, can the other admin determine what my DES password is?
Not easily. This would amount to a known-plaintext attack on DES -- the "attacker" has access to a pair of ciphertext and plaintext, and is attempting to determine the key. DES is not a particularly strong cipher, but it is not considered to be especially vulnerable to this attack.
It really look into the money that one can spend on it. For example, the Copacobana from 2007:
With further optimization of our implementation, we could achieve a clock frequency of 136MHz for the brute force attack with COPACOBANA. Now, the average search time for a single DES key is less than a week, precisely 6.4 days. The worst case for the search has been reduced to 12.8 days now.
Recently there is another one crach.sh
we’ve built a system with 48 Virtex-6 LX240Ts which can exhaust the keyspace in around 26 hours, and have provided it for the research community to use. Our hope is that this will better demonstrate the insecurity of DES and move people to adopt more secure modern encryption standards.
They can also find the key in 25 seconds with a prebuilt rainbow table built for a predetermined plaintext-ciphertext pair.