A very good question. An H-1B petition allows for full or part-time work so long as you are being compensated above the Prevailing Wage, and complying with the other general H-1B rules and regulations.
However any change in your H-1B will require your employer to submit an amended H-1B petition, prior to any changes in your current approved H-1B.
You may attend classes part-time so long as it is not the primary purpose of your stay in the US, and merely incidental to your primary work on the H-1B.
Your other options could involve attending a school that allows for CPT work as part of your degree program. To pursue this you would change your status to F-1 Student, enroll in school full-time, and work on CPT as part of your degree. After you graduate you can work on OPT for up to 12 months, and if your major is in STEM you can extend your F-1 OPT for a further 17 months, all before going back on H-1B.
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Most likely not as the H-1B visa implies that you are working full time for your employer. Consult with an experienced attorney who can give you a better advise based on the circumstances and details of your case.