Burnley have won just three of their opening 16 games and sit 19th in the table, six points from safety.
"I feel the pressure every time I'm sat in this position, I've felt the pressure every time I've walked out on the football pitch for 20 years of my career," said Parker.
"There's no-one will ever, no owner, no fan, yourself asking me the question or the guy who's asked you to ask the question to me, that will put more pressure on me than I put on myself.
"I'm a winner, I'm a fighter and throughout my career and throughout my life, that's exactly what I've been, and there's always been pressure. That's just the way it is.
"Pressure is something you learn to deal with, and also it's that pressure and that fear at times - because no doubt at times there's fear - that drives, absolutely drives."