How to approach deep learning CV/Resume parser using Convolutions?
Typically, resumes are not images. Almost all resumes are Microsoft Word or pdf. Given those document formats, a deep learning parser should be sequence-based (e.g., Recurrent Neural Network (RNN) or Long Short Term Memory (LSTM)). 

To apply Deep Learning, you'll need many thousands of examples with each section labeled. There is HR-XML (Human Resources - Extensible Markup Language) which are the industry standards for labels of resume sections.