It seems to me that 1 and 2 are correct.
All three sentences are correct, but the punctuation of (1) needs improvement. Placing the "in"-phrase right after the subject makes it a parenthetical, and parentheticals are set off by commas or parentheses in writing:
(1a) James Parker, in his book “Over the Rainbow,” suggested that a successful business has the following characteristics.
Sentences (2) and (3) work equally well. The "in"-phrase is an adverbial, and adverbials are mobile: they can be placed in many different positions. Where the "in"-phrase would NOT work in your example is at the end of the sentence:
4. ?? James Parker suggested that a successful business has the following characteristics in his book “Over the Rainbow."
Although (4) is not ungrammatical, it gives readers a miscue. At first sight, (4) appears to be saying that the characteristics, not Parker's suggestion, are in "Over the Rainbow." If you put the reference at the end, make it academic:
5. James Parker suggested that a successful business has the following characteristics ("Over the Rainbow," p. 123).
Lastly, it would be good to change "suggested" to "suggests," because you are talking about an author's view, and authors' views are generally reported in the present tense. Also, italics may be used for the title.