Ignition optionally interprets but always parses javascript, optimizes and compiles bytecode. So IMO Ignition is primarily a compiler frontend that has an optional interpreter back end built in.
I could see them splitting Ignition into a bytecode compiler and separate interpreter and give them separate names.
The V8 parser predates Ignition, so most V8 developers will consider it a separate subsystem and Ignition really to be just the bytecode format and interpreter for that bytecode. The parser was used to feed fullcodegen back in the day, and TurboFan also used to have an AST-to-TurboFan phase. Nowadays, I believe the only JavaScript path into TurboFan is actually through Ignition bytecode. TurboFan also has a frontend for Wasm bytecode (or viewed conversely, the Wasm engine has a TurboFan IR generator, since this lives in the wasm directory and is maintained by a subteam).
So does the parser generate byte code or ignition? Is ignition just an interpreter or is it a byte code generator too? Does ignition take byte code or is it fed the ast from the parser to generate byte code?
"Ignition" was the codename for the entire interpreter project, including both the Ignition compiler and the Ignition interpreter. So the answer to your question is "yes".
The parser produces an AST, which is an in-memory data structure that is a tree. Ignition (in src/interpreter) has a visitor that walks over the AST and generates bytecode.