Here is a list of some programming languages that are not written in C, alongside the languages they are implemented in:
- Haskell - Haskell
- Idris - Haskell
- Adga - Haskell
- PureScript - Haskell
- Elm - Haskell
- Mercury - Mercury
- Rust - Rust (initially OCaml)
- Go - Go
- Crystal - Crystal
- OCaml - OCaml
- Frege - Frege + Java
- Haxe - OCaml + Haxe
- Scala - Scala
- Futhark - Haskell
- ATS - ATS
The best languages for implementing a compiler are probably going to be pretty far away from C. Functional languages give you things like recursion schemes and monadic parser combinators (provided you have typeclasses), which makes them especially suited for compiler work.
Second, to address your question as to whether C is the "mother/father of all programming languages" - not really. C was an incredibly well-designed language at the time it appeared, and it no doubt has influenced language designers who then went on to do very different things. But at the end of the day, Haskell departs from C in essentially every way possible. C is 45 years old and it is not surprising that we have learned to do better in the meantime.
Finally, to answer your third question, it is simply not the case that C implements "all the concepts". In particular, trying to implement some of the advanced concepts from functional programming (such as metamorphisms, or, god forbid, synchromorphisms) in C would be monstrously difficult. I am not particularly familiar with object-oriented programming, but I do know for a fact that some object-oriented languages have sum types.