The amount of alcohol absorbed into the pickle would be so miniscule it will likely cause no harm. First, beer is relatively low in alcohol as is. You are then diluting it with vinegar and boiling some off. It gets diluted further by the water from the vegetable you are pickling. First the water is drawn out by osmosis from the brine. As salt gets absorbed into the vegetable and the brine is diluted by the leached water, reverse osmosis occurs, and a little of the brining liquid is drawn into the vegetable. This will only have a trace of alcohol if the brine is made with beer.
A lot of everyday foods have trace amounts of alcohol like orange juice (or any natural juice), yogurt, even rye bread. "Non Alcohol" beer can have up to .5% alcohol. Wine vinegars can have as much as 2%. The probiotic bacteria and yeasts in our intestines put trace amounts of alcohol in out bloodstream.
Let your father enjoy your pickles. Unless he is into guzzling pickle brine he will be fine.