Depending on how flexible you are and can be, there are many opportunities for folks that are during "off hours" especially if the research is more community-based. We do some of our interviews on weekends and late evenings (if its a home visit, and not done at the child's school), and a few of my RAs only come in during the early evening/weekends to do data-entry.
The only roadblock might be lab meetings which are typically held on a weekday between 9am - 5pm, and depending on the PI, they might want you to be there in person, particularly when you are first starting. I also find that you will have a much richer research experience if you work during the times where you can interface with other RAs, grad students, postdocs, PIs, etc. You might have to be flexible and leave work for 1-2 hours per week to join the lab meeting. Is that a possibility, for example, can you make that your lunch hour? A requirement of being in our lab is that you must attend our weekly lab meeting in person, although we are willing to waive this once an RA has established themselves in the lab (for example, if they take on a summer position, we might not require them to attend the meeting every week and instead do a check in during the summer).
In sum, I think if you are open to what experience you can get, just ask around and see what opportunities there may be. I wouldn't drop down to part-time work to get research experience in someone's lab. If anything, try to find paid experiences (e.g., project coordinator), they are out there although they are very competitive, and many times you might be competing with Master's level folks for the position.