^You automatically have parental responsibility if you’re the child’s mother, but you still need the permission of anyone else with parental responsibility before you take the child abroad.
You can take a child abroad for 28 days without getting permission if a child arrangement order says the child must live with you, unless a court order says you can’t.
Get permission from someone with parental responsibility
A letter from the person with parental responsibility for the child is usually enough to show you’ve got permission to take them abroad.
You might be asked for the letter at a UK or foreign border, or if there’s a dispute about taking a child abroad. The letter should include the other person’s contact details and details about the trip.
It also helps if you’ve:
evidence of your relationship with the child, eg a birth or adoption certificate
a divorce or marriage certificate, if you are a single parent but your family name is different from the child’s^
So, a letter. That shouldn’t be difficult to forge. 
And “contact details.” A postal address counts, right?
Anyone planning to abduct a child could easily think this through and go as far as to have a burn phone with a recorded answering message with any male (or female) voice to verify the “identity” of the other parent. “Oh, he’s probably in the plane now, that’s why we’re having a nice break while he’s gone.”
It’s a very bad system altogether. Does zilch to protect children.
I took my husband’s surname because I didn’t love my own (plus it was my father’s, not necessarily mine), it greases the social and admin wheels (I know, weak but true, I’d hate the faff of explaining), and—I suppose—I’d always grown up with an unspoken expectation to do so. But really it can down to avoidance of admin. 
I can’t care much about it now