A Little Bird and a Monkey Butt
Yes, STD is funded by an insurance policy, I am pretty sure the premium gets pulled after taxes, only my health insurance and dependent care are pre tax.
Then no, your STD payments should not be taxed.
When planning, make sure you double check your plan's exclusion/waiting periods. My plan administrator told me that I could expect 6 weeks of pay for a V-birth, but it was really only 5 weeks because they had a 1 week waiting period.
Congrats on the BFP!