I'm not the most outgoing person either but often times it's the caring attitude and attentive listening that is more important to parents than constant babbling and incessant small talk. I mean, their baby is in hospital and they are not there to make friends with you. That being said, some small talk can help parents feel more at ease with you and their environment.
I usually try and focus on the baby. If you've just met them and baby is in a stable condition, you might ask (if baby has an unique name or something) how they came up with that name, have siblings met baby yet (obviously first make sure they haven't had a twin loss or something), if they live far and how their commute is going, etc. If I've had baby a couple of times, I might say oh they've gotten bigger since I saw him last, he seems to really like his soother, he looks like he's going to have your hair colour. You'll find parents are happy to go on about their babies at length.
Also, try and converse while you're doing something. It makes like less awkward and less like you've rehearsed a speech. Something like when you're changing a diaper or taking a temperature, that gives you something to do but doesn't require your full attention like bloodwork or ETT suctioning. Also involve parents more in the care and give them teaching tips. That way you won't feel like they're staring at you waiting to fill the silence.
Hope this helps. From one introvert to another. :)