Thanks to all for swift reply!
To answer all questions I obtained from you:
William (° 31/8/1842) must have emigrated to Antwerp between 1860 and 1865. He married a local girl, ML Parmentier at Antwerp on 28/11/1868. His marriage records mentions his parents as William Bethell & Mary McCoy, both “deceased” at that date. (Which does not guarantee they died before that date..?) He was active in ship managing over here and I presume he sometimes got back to Liverpool, it is certain he died there, because his death card says so. Back in 2000, I obtained photographs of Yew Tree Cemetary tomb stone which confirms this.
I do not know whether the parents were really married, if there is no record to be found, maybe there were not.. In any case, there should be a birth record of William jr. It looks to me that the one mentioned by CaroleW might be the one. I checked several censusses some years ago, and found a William ° 1827 and a Mary ° 1828.. but nothing to give certainty. Thank you so far.