The signatories of a child’s deed poll has nothing to do with parental responsibility and everything to do with birth, in almost all circumstances. I.e. you usually need the birth mother and biological father’s signatures, OR the legally adoptive parent(s) signatures, regardless of who actually cares for, has parental responsibility for, or has custody over the child. There are exceptions, but they’re even more complicated.
In short, I’d only recommend making your own deed poll for a child if you’re very confident that you’re a “simple” case. Otherwise, speak to a solicitor or a specialist service. Get multiple quotes.