Since Rust is a small village, there's no chain hotels here as the majority of properties are guesthouses with around 10 rooms. Your nearest 24 hour reception hotel (aside from the theme park hotels as Diogo mentions) will be the Highway Hotel in Herbolzheim. It's located in the service station a junction down from Europa-Park. It's....ok, but nothing to shout home about. Just your bog standard motel really.
For chain hotels, you're going to need to head around 30-40 minutes south to Freiburg, where there's plenty of chains such as Novotel, Mercure, Intercity etc.
Whilst I know stepping away from the "you know what you geting" chains can seem a bit daunting, demand for accommodation in Rust is massive - which has thankfully meant that everything tends to be pretty decent quality. I've never stayed anywhere which hasn't been decent in Rust, and it's nice to be in walking distance of the theme park and hotel bars. You're not going to get everything you want at a guesthouse or hotel in Rust, but hopefully there's a few of your concerns that can be addressed:
Free Parking - Almost all guesthouses in Rust have free parking for guests. You'll even usually get one day's free parking at Europa-Park itself when checking in - very useful if you want to pop on park after checking out.
24hr reception - As mentioned, there's no 24 hour receptions at guesthouses in Rust. That said, owners are often willing to be flexible if you let them know beforehand. One in particular is one of my favourites - Elzpark, which has a handy lock box out the back. If you're going to arrive late, let them know and they'll issue a code to get into the lock box and collect your key.
No locked doors - As above, you can often arrange something to get in if there is no reception. In any case, owners are on site each morning for 3-4 hours and then again on the afternoon at an absolute minimum. There are also emergency contact numbers either posted on the door and/or in the welcome packs in your room/apartment should you need to get hold of anyone. Many either live in an apartment on site, or are just down the road in a separate home.
Multi Lingual Staff - Due to the number of visitors who are non German or French speakers, I've never encountered this as an issue, especially in recent years. However, if you'd like reassurance, sites like booking.com state "we speak your language!" in the description if a guesthouse/hotel has confirmed they have English speaking staff.
Of course, if you're willing to splash out (and do do this once if you can!) there's always the official hotels which cover all of the above with no issue at all. But if you're budget conscious, don't be put off by staying in an independent hotel