Two separate acts essentially combine for income replacement and job protection. CTPL is for income replacement, and CTFMLA for job protection (and unpaid leave entitlement). To my knowledge you can use both concurrently. I would reach out to DOL to get info on the latter.

https://portal.ct.gov/DOLUI/FMLA-FAQs

The important section:

Pay While on Leave

Q9. Is my employer required to pay me when I take CTFMLA leave? Generally, CTFMLA leave is unpaid. However, an employee’s accrued, paid leave time with the employer, such as vacation, sick leave, personal leave or paid time off, may be applied to the leave if required by the employer or requested by the employee. An employee may choose to preserve up to 2 weeks of their accrued, paid leave time.

Additionally, wage replacement benefits under Connecticut’s Paid Leave program may apply, as well as short-term or long-term disability benefits, or workers’ compensation depending upon the eligibility requirement for those leaves. More information about Connecticut’s Paid Leave program is available at https://ctpaidleave.org/.