The correct answer is:
As they gazed at their new grandbaby, Shelby's parents' excitement was plain to see.
The first option is the correct grammar.
The second option doesn't uses the comma to connect both sentences, thus is incorrect,
The third option uses "parent's" when the correct grammar is "parents'" because it is plural, so the apostrophe goes after the "s".
The fourth option uses "Shelbys'" as it is a pplural noun, Shelby is a singular person, thus the apostrophe goes before the "s" as in Shelby's.
