One toddler takes away the mouth piece of the other which makes him cry.
A toddler has another kids mouth piece.
There is no indication of ownership of the mouth piece in the first sentence.  There is no indication that the mouth piece is another kids.
exists e1 e2 s1 x1 x2 x3 x4 x5 x6.(n_male(x2) & n_other(x4) & n_piece(x2) & of(x2,x3) & n_mouth(x3) & a_away(s1) & Manner(e2,s1) & Theme(e2,x2) & Actor(e2,x1) & v_take(e2) & Recipient(e1,x2) & Theme(e1,x6) & Actor(e1,x5) & v_make(e1) & n_cry(x6) & (x2 = x5) & of(x2,x4) & n_toddler(x1) & card_eq_1(x1))
exists e1 x1 x2 x3 x4 x5 x6.(Theme(e1,x5) & Actor(e1,x1) & v_have(e1) & n_piece(x5) & of(x5,x6) & n_mouth(x6) & of(x5,x3) & n_kid(x3) & -(x5 = x2) & n_piece(x2) & of(x2,x4) & n_mouth(x4) & of(x2,x3) & n_toddler(x1))