Parents' joy doesn't show on the outside. The same goes for their sadness and anxiety.
Literal
Parents [possessive-の] joy [topic-は] outside [to-に] doesn't-appear. That sadness [and-や] anxiety [also-も] also so.
しかり (also so / likewise) is a classical/formal expression meaning 'the same applies.' や lists items non-exhaustively ('sadness and anxiety, among others'). The sentence makes a general observation about parents concealing emotions — culturally resonant in Japanese society where emotional restraint is valued.