For a language that has entirely trivial syntax, just splitting by spaces is a parser, and that’s fine. Concatenative languages with only number literals can be one case of that, for example.
It doesn’t look like your language is quite that, however: your example contains a space within the first quoted string! Splitting by spaces will break that apart, and while you can repair it, the work to do so and future-proofing would point me towards making a simple parser, personally. Here the parser really is essentially a lexer, but it will address keeping quoted strings as single terms, perhaps escapes, etc.
If you’re going to have even slightly more complex syntax — like quotes or function literals — you’ll probably need a real parser. Those are very common to enable things like conditionals; sometimes a “jump to last [“ suffices, but only in the very simple cases.
Any sort of structure or nesting brings that into play, so while for the very early stages of implementation you can probably get away without one, it’s likely you’ll have to make one eventually. If the language really is just a linear front-to-back model perhaps you escape that.