Are OCONUS assignments as a single parent doable? Are there any overseas assignments that would be considered single parent friendly?

I spent 6 years in Germany and the next 3 1/2 in Hawaii. Deployed twice out of Germany. Yes, it is doable. I recommend you have several layers of a FCP for short and long term with a detailed LOI. I recommend this regardless of place of duty...you just never know what will happen. Also...know the regulation!
I've been a single mom in the Army for 12 years and am currently on my 3rd deployment away from my child. I understand the pains of the short term provider, especially overseas when you are totally dependent on fellow military members. When looking into child care, the in home care providers do often provide overnight for up to a certain number of days. This is reliable as the homes are regularly inspected and certified. Plus you can interview them as well. A great option for field exercises. Also make sure you get the NEO passports for all dependents.
As a commander, I had 22 single or dual military families (FCPs). I only had problems with one single parent.

Most OCONUS requires that single service members and dual military be deployment ready within a short period following PCS move (a Family Care Plan). This is something you should research very carefully before seeking an oversees assignment. Most organizations in Europe and Asia deploy, only MAACOM HQ's do not deploy as a command, but they too have individual replacement requirements. They also love to train in the field for extended periods, not having an extended care sitter is no excuse for missing deployments or training. It can be done, just be sure it is right for you. It is also possible that you could get assigned oversees and meet the same challenges.

After reading through the three previous responses, I have determined there really isn't anything to add. As a Corporal, you have a higher chance for a deployment than others, so it really does it matter if your next assignment is CONUS or OCONUS; you will need a family care plan.