I'm a single mum of teenagers - oldest kid is approaching adulthood - and would suggest that Penny's mum would most likely do everything she could to hide how she was feeling to Penny, not wanting her to miss such an important opportunity. (but of course she is your character and not everyone will do this)
In her mum's shoes I would:
1. let my child know that I will miss them, and encourage them to visit and call/text often, and keep me updated with what's going on in their life - I'd frame this as "I'm still your mum and always going to care about what you're doing and that you're happy and want to hear about your achievements" and not "please stay" (to be honest I'd put aside any gut instincts of wanting my kid to stay as I'd want them to be a happy, independent adult more)
2. probably still cry when my kid leaves, and try to hide it, like you try to hide your tears on their first day of school so as not to freak them out. Not much different really... just the kid is that much bigger and they're going away for a bit longer
3. make sure my kid knows they'll always have a place to stay at my home, no matter what else happens. (albeit I'd charge them keep unless there was some kind of emergency going on where they weren't able to work for the time being)
I also think that Penny's mum should start planning to do non child related things and expand her social circle so she doesn't feel quite so lonely with Penny gone. But of course she's your character, not me.
Things I wouldn't do but Penny's mum might, if she really doesn't want Penny to go away:
1. Try to convince Penny that the local college courses, teachers, opportunities etc are just as good as where Penny's going
2. Keep talking about how lonely she'll be when Penny's not there
3. Suggest career options that would keep her living at home
4. Suggest how after going to college (if she really can't stop Penny going) she could come back to this town and live with her mum again and pursue her career after college while living with her mum
5. Point out how expensive rents and mortgage costs are, and how she'll be able to save lots of money by living at home. Maybe she's the kind of mum that wouldn't charge her kid keep as a way to encourage her kid to not leave
6. Worry out loud about things that Penny might have to face living on her own - dangers, difficulties with cost of living, day to day chores, doing everything by herself - she could give off a vibe of not believing that Penny would be able to cope with those things without her mum around to help her. In a sense, infantilising her. It can be hard as parents to cope with how fast kids grow up and even when they're legally adults, they still at some level feel like your little babies. It can take an active effort to let them go and do age appropriate things on their own. I was raised to be independent from a young age and I've raised my kids the same, but not all parents have that mentality and that can make it even harder to let your little babies go and to grown up things all by themselves.
It's not clear if Penny's mum's actively going to try to manipulate Penny into staying, or if you just want to show Penny's mum's true feelings while she's encouraging Penny to follow her heart and life goals. Some of the things I've suggested might be too manipulative for what you're aiming for. But hopefully it can give you some ideas or somewhere to start.