OK, I'll bite. I am local, used to live on the outskirts of Cardiff but now near to Llantwit Major which is 10 miles west of Cardiff.
Personally, I wouldn't feel comfortable stopping at Adam St NCP. It's a rough area. Marginally better is Sophia Gardens car park but if you are coming for an event (Taylor Swift fan, perhaps?) or there is an event on, don't even think about entering the city centre. It will be rammed.
If you like pub stops, I recommend the Plymouth Arms at St Fagans which has a large car park at the rear. There are regular buses to the centre and you can walk to the Museum of Welsh Life - well worth a visit and free. Another good pub stop is the Blue Anchor at Aberthaw.
If you need services there are two sites near Llantwit Major - Acorn and Happy Jakes. I haven't stayed there because they are on my front door but there is plenty of parking at the local rail station (20 mins to the centre, every hour)
(Since I've been typing this your post has appeared that you would prefer to walk into the centre. Knowing how rough Cardiff centre has become over the last few years, I really wouldn't overnight there, but that's just me)