I left mine at home with my husband when my eldest had a birthday party to go to. It was just easier!
For us it depends on whose birthday it is. If it is a family member or close friend (e.g we have met partners or spent time together with partners/kids) then I would go as a family. If it's a school friend or someone who hasn't met my husband/second child then I wouldn't bring the second child. Or if the parent hasn't mentioned to bring my second child then I generally wouldn't. Also if it's softplay they usually charge per person, so if the birthday child's family have paid for a package/set amount of children, it's unlikely the sibling would be included. As you have a baby who wouldn't be involved in the activity I would assume this wouldn't be an issue but maybe an issue if the place is small/have been allocated a certain number of seats -So I would ask if not sure ...
My partner works at weekends so if we are invited to a party when I'm on my own with both I just send the parent a quick message to ask if it's OK I bring him along and everyone has always been happy for him to come too.
My 3yo went to her first birthday party a few months ago and I didn’t even think to ask if I could bring my 8 month old. I would feel quite offended if he wasn’t allowed to be with me tbh as he didn’t eat any of the food or crawl around getting into trouble. If he did do those things then I probably wouldn’t have brought him …
I’d imagine it’s usually fine when they are so young but if in doubt I’d message / speak with the organiser and check it’s ok to bring them along