Part time internship positions are hard to come by unless you are willing to work unpaid. For companies, it's risky to hire interns, too. Sometimes they get very good people, and sometimes they get people like me. But in all cases, the company will have to spend significant money to train you (this applies to full time positions, too), and there's always a chance that they won't get the value back from your work. The chance of them getting the value back is even less if you can only work part time.
From my experience (I've done 3 internships and applied to about 20), many companies won't even consider 4 month internships, because that's about the amount of time they need to get a new hire (intern or full time) up to speed so they can start contributing significantly. 8 months is about break even for them, and 4 months is almost always a net loss for them, even if you work unpaid, because their cost to you is not just your salary, but also the salary paid to senior employees that have to spend time to train you, instead of doing other useful work.
