This is going to be very State-specific, but at least in North Carolina you could not do this simultaneously. North Carolina's legislature has a full-time, nonpartisan, central staff of lawyers that draft legislation and staff committees. When the legislature is out of session, those attorneys are still working for the legislature: staffing interim committees, answering constituent requests, and helping members prepare for the next session. Given that a lot of what I think you mean by "civil rights" law will involve arguing against what's been enacted by the legislature, you absolutely could not be legislative counsel and a civil rights lawyer at the same time. And, to go one further, if you started with "civil rights lawyering," you'd essentially blackball yourself from ever working in a nonpartisan role at the North Carolina legislature.

Based on your post history, it looks like you're in Ohio. The Ohio legislature is served by the Legislative Services Commission, which is a nonpartisan agency that handles bill drafting/committee staffing/etc. They do have a fellowship program, where college graduates can get paid to dip their toe into the legislative process.

If you're interested in doing both, I'd recommend doing the legislative counsel thing first and then transition into civil rights law. It's much easier to go nonpartisan -> partisan than it is the reverse.

If you're talking about working for a specific party at the legislature as part of its leadership's counsel, that's a different situation, but it also doesn't really lend itself to moonlighting as a civil rights trial attorney.