The MP spoke out during a Westminster debate on supporting fathers in early parenthood.
Mr Bradley discussed his experiences as a new father, like not being allowed to stay with his new baby in hospital outside of visiting hours, and the difficult emotions he experienced watching his wife bond with his son.
The MP brought up issues that many new fathers face, and the importance of ensuring that support is available, especially for working class and low-income fathers where finances add additional pressure.
Mr Bradley said: “It was good to have the opportunity to discuss early parenthood and support for fathers in Parliament. In my speech I recalled some of my own experiences as a young father of two boys.
"The debate was not to detract from the challenges of motherhood in any way but offered a chance to highlight the challenges and barriers that also exist for men which are often overlooked.
"I think sometimes men are reluctant to talk about the emotions they feel in these situations and often feel like a bit of a ‘spare-part’ in those early days as attention is quite rightly focused on new mothers.
"It is equally important though to highlight the challenges faced by fathers and offer support.
"We know that young men have the highest rates of mental health challenges, partly because they don’t tend to talk about it, and we know that children are best supported with a male role model in their lives.
"We need the services to back that up.”
He also asked the under-secretary of state at the department of health, Jackie Doyle-Price MP, to look at how barriers can be reduced between social services, health care providers and local authorities to ensure a joined-up approach and efficient use of funding.
Ms Doyle spoke about the government’s work to invest in facilities in maternity hospitals to ensure that fathers can stay in post-natal units.
She also highlighted the importance of ensuring that health visitors look to support fathers as part of their regular visits and contact in the early years of a child’s life.