ばかりでなく

Not just X but also Y (Written)
Plain form verb/Noun/い Adjective + ばかり＋でなく
な Adjective + なばかり＋でなく

ばかりでなく is simply a combination of ばかり + でなく (ではない）.
In other words, “Not only/Not Just”.
There should be a “also” or “even” meaning word in the sentence after it, such as も、まで、or さえ。
ばかりでなく tends to be written more than spoken.

Examples

• 彼ばかりでなく、皆もそうお思っていた
  It wasn’t just him, everyone was thinking it.
  
  
• 信じたばかりでなく、金まであげた
  I didn’t just believe him, I even gave him money.
  
  
• 優しいばかりでなく、気前もいい
  Not just nice, but also generous
  
  
• 穏やかなばかりでなく、家賃も安い
  Not just peaceful, but the rent is also cheap.
  
  
• 日本語ばかりでなく、英語さえもわからない
  Not just Japanese, he doesn’t even understand English either. 
  
  

だけでなく

Not only X but Y (Spoken)
Plain form verb/Noun/い Adjective + だけ＋でなく
な Adjective + なだけ＋でなく

だけでなく is more or less interchangeable with ばかりでなく.
ばかりでなく tends to be written Japanese more than spoken,
While だけでなく tends to be spoken.

Examples

• 穏やかなばかりでなく、家賃も安い
  Not just peaceful, but the rent is also cheap.
  
  
• 穏やかなだけでなく、家賃も安い
  Not only peaceful, but the rent is also cheap.

• 彼ばかりでなく、皆もそうお思っていた
  It wasn’t just him, everyone was thinking it.
  
  
• 彼だけでなく、皆もそうお思っていた
  It wasn’t just him, everyone was thinking it.
  
  

ばかりか

(I can’t believe it) Not just X but Y
Plain form verb/Noun/い Adjective + ばかりか
な Adjective + なばかりか

ばかりか is another variation of ばかりでなく. Its stronger and more formal, and emphasizes emotion more than ばかりでなく.
Typically this emotion is shock or anger.

Examples

• ミスしたばかりか、嘘もついた
  Not only did he make a mistake, he lied about it. (Angry). 
  
  
• 親を殴ったばかりか、財布も盗んだ
  He didn’t just hit his parent, he stole their wallet.
  
  
• 彼は優しいばかりか、自分の食べ物さえ他人にあげる
  He isn’t just nice, he even gives his own food to other people.
  
  
• アホなばかりか、気づきさえもしていない。
  He isn’t just stupid, he doesn’t realize it either.
  
  
• 持ってきたばかりか、飾りもしてくれた
  He didn’t just bring it, he also decorated it for us. 
  
  

References

HiNative (Difference between だけでなく・ばかりでなく）