I haven't made the Show HN post yet, but using the parser combinator library that I've been building[1], here's an answer to your exercise:

  module Joker_vDChallengeParser
    include Parsby::Combinators
    extend self

    def parse(s)
      (expr < eof).parse(s)
    end

    define_combinator(:exp_op) {|left, right| group(left, spaced(lit("^")), right) }

    define_combinator(:pos_op) {|subexpr| group(lit("+") < ws, subexpr) }
    define_combinator(:neg_op) {|subexpr| group(lit("-") < ws, subexpr) }

    define_combinator(:div_op) {|left, right| group(left, spaced(lit("/")), right) }
    define_combinator(:mul_op) {|left, right| group(left, spaced(lit("*")), right) }

    define_combinator(:add_op) {|left, right| group(left, spaced(lit("+")), right) }
    define_combinator(:sub_op) {|left, right| group(left, spaced(lit("-")), right) }

    define_combinator(:call_op) {|left, right| group(left, ws_1, right) }

    define_combinator :identifier do
      first_char = char_in([*('a'..'z'), *('A'..'Z'), '_'].join)
      rest_char = first_char | char_in([*('0'..'9')].join)
      first_char + join(many(rest_char))
    end

    define_combinator :lambda_expr do
      group(
        sep_by_1(ws_1, identifier) < spaced(lit(".")),
        expr,
      )
    end

    def scope(x, &b)
      b.call x
    end

    define_combinator :expr do
      lazy do
        e = choice(
          decimal,
          lambda_expr,
          identifier,
          between(lit("("), lit(")"), expr),
        )

        # Each e = scope ... block is a precedence level. You can switch
        # them around to play with the precedence of the operators.
        #
        # hpe -- higher precedence level expression
        # spe -- same precedence level expression

        # Basing myself on Haskell and making function calls the highest
        # precedence operators.
        e = scope e do |hpe|
          reduce hpe do |left_result|
            choice(
              call_op(pure(left_result), hpe),
            )
          end
        end

        # Our only right-associative precedence level.
        e = scope e do |hpe|
          recursive do |spe|
            choice(
              exp_op(hpe, spe),
              hpe,
            )
          end
        end

        e = scope e do |hpe|
          recursive do |spe|
            choice(
              neg_op(spe),
              pos_op(spe),
              hpe,
            )
          end
        end

        # Left-associativity done by parsing left operand bottom-up and
        # right operands top-down.
        e = scope e do |hpe|
          reduce hpe do |left_result|
            choice(
              mul_op(pure(left_result), hpe),
              div_op(pure(left_result), hpe),
            )
          end
        end

        e = scope e do |hpe|
          reduce hpe do |left_result|
            choice(
              add_op(pure(left_result), hpe),
              sub_op(pure(left_result), hpe),
            )
          end
        end
      end
    end
  end

That was a nice exercise. Here's some example parses:

  pry(main)> Joker_vDChallengeParser.parse "- 3 - foo bar . foo - bar - 5"                                     
  => [["-", 3], "-", [["foo", "bar"], [["foo", "-", "bar"], "-", 5]]]
  pry(main)> Joker_vDChallengeParser.parse "- 3 - (foo bar . foo - bar - 5) - 4 * -2 + 5 / 2"                 
  => [[[["-", 3], "-", [["foo", "bar"], [["foo", "-", "bar"], "-", 5]]], "-", [4, "*", ["-", 2]]], "+", [5, "/", 2]]
  pry(main)> Joker_vDChallengeParser.parse "- 3 - (foo bar . foo 5 6 - bar - 5) - 4 * -2 + 5 / 2"             
  => [[[["-", 3], "-", [["foo", "bar"], [[[["foo", " ", 5], " ", 6], "-", "bar"], "-", 5]]], "-", [4, "*", ["-", 2]]], "+", [5, "/", 2]]

  pry(main)> Joker_vDChallengeParser.parse "foo bar . foo bar baz qux . baz qux"                              
  => [["foo", "bar"], [["foo", "bar", "baz", "qux"], ["baz", " ", "qux"]]]
  pry(main)> Joker_vDChallengeParser.parse "foo bar . foo bar + baz qux . baz qux"                            
  => [["foo", "bar"], [["foo", " ", "bar"], "+", [["baz", "qux"], ["baz", " ", "qux"]]]]

You can try it by cloning my repo, running `./bin/console`, and pasting the module, or putting it in a file in `lib/`. If you put it in a file, you can reload changes with `reload!` in the console.

[1] https://github.com/jolmg/parsby

EDIT: Fixed the syntax for newer versions of Ruby.