The sum of some parts

05AB1E, 10 bytes

æʒø`O²‹_i,

Try it online!

Explanation

æ           # compute powerset of input
 ʒ          # filter out falsy values (we don't actually filter, it just saves a byte)
  ø         # zip into a list of keys and a list of values
   `        # push separately to the stack
    O       # sum values
     ²‹_i   # if not less than input_2
         ,  # print list of keys

Mathematica, 45 43 bytes

^{Saved 2 bytes due to alephalpha.}

Keys/@Cases[Subsets@#,a_/;Tr@Values@a>=#2]&

Anonymous function. Takes a list of rules (e.g., {"Part 1"->20,"Part 2"->15}) and a number as input and returns a list of lists of strings as output.

Japt, 15 14 12 bytes

Saved 2 bytes thanks to @obarakon

à f_xŨVÃm®g

Test it online! Output is a nested array, but it's rather hard to tell so the -R flag has been added to put each item on its own line.

Explanation

 à f_  xÅ  ¨ VÃ m®   g
Uà fZ{Zxs1 >=V} mmZ{Zg}
                         // Implicit: U = input array, V = input number
Uà                       // Take all combinations of U.
   fZ{        }          // Filter to only the combinations Z where this is true:
      Zxs1               //   The sum of each [item of Z]'s item at index 1
           >=V           //   is greater than or equal to V.
                m        // Map each combination in the result by
                 mZ{  }  //   mapping each array Z in that to
                    Zg   //     the first item in Z.
                         // This replaces each key-value pair with its key.
                         // Implicit: output result of last expression

An interesting thing to note is the use of Å (or s1 ) instead of g1 . s1 on an array simply returns everything but the first item. Since each array only has two items (a key and a value), this returns an array containing the value. x then uses JS's native coercion to convert this into a number, and on an array containing one number, this happens to return that number.