Split json file as parent and multiple child

Regarding @Dmitry's points (that [ "$ref": "book1.json" ] is not valid JSON), why don't you instead have

[
  { "$ref": "book1.json" },
  { "$ref": "book2.json" },
  ...
]

Beyond this, what you're asking for is a program that generates multiple files, one of which is Parent.json and the rest of which are bookn.json. Since jq can only provide output via one stream, you could generate one line of compressed JSON per output file, the first line being Parent.json, and the remaining ones each bookn.json. You'd then have to post-process each line and pipe it into the appropriate file, and you could uncompress the line at that point.

1. Producing those multiple lines,

$ jq -c '.store.book as $books | .store.book = [
      { "$ref": "book\(range(1; $books | length + 1)).json" } ], $books[]' Main.json
{"store":{"book":[{"$ref":"book1.json"},{"$ref":"book2.json"}],"bicycle":{"color":"red","price":19.95}},"expensive":10}
{"category":"reference","author":"Nigel Rees","title":"Sayings of the Century","price":8.95}
{"category":"fiction","author":"Evelyn Waugh","title":"Sword of Honour","price":12.99}

2. Piping those lines into each their file, uncompressing each one,

$ jq -c '...' Main.json | (
      IFS= read -r parent
      jq . <<< "$parent" > Parent.json

      i=1
      while IFS= read -r book
      do
        jq . <<< "$book" > "book$i.json"
        i=$(($i+1))
      done
  )

My sh is a bit rusty, storing reads seems unnecessarily complicated.

Update: In the spirit of Thor's excellent solution where he splits the tasks of generating Parent.json and bookn.json into two tasks, and solves the second part with split which seems pretty cut for it, I thought I'd see if I could do something similar:

genbooks() {
  jq -c '.store.book[]' |
  perl -nlE '$i++; open my $fh, "|jq . >book$i.json"; say $fh $_; close $fh'
}

You could do it all in one go with jq and coreutils, e.g. define two functions like this:

genparent() {
  jq '.store.book = [ { "$ref": "book\(range(1; .store.book | length + 1)).json" } ]' > Parent.json
}

genbooks() {
  jq -c '.store.book[]'           |
  split --numeric-suffix=1        \
        --suffix-length=1         \
        --additional-suffix=.json \
        --filter='jq . > $FILE'   \
        --lines=1 - book
}

And run it like this:

<Main.json tee >(genparent) >(genbooks) > /dev/null
ls

Output:

book1.json  book2.json  Main.json  Parent.json

Parent.json

{
  "store": {
    "book": [
      {
        "$ref": "book1.json"
      },
      {
        "$ref": "book2.json"
      }
    ],
    "bicycle": {
      "color": "red",
      "price": 19.95
    }
  },
  "expensive": 10
}

book1.json

{
  "category": "reference",
  "author": "Nigel Rees",
  "title": "Sayings of the Century",
  "price": 8.95
}

book2.json

{
  "category": "fiction",
  "author": "Evelyn Waugh",
  "title": "Sword of Honour",
  "price": 12.99
}