Cycles lanes come in two falvours. Advisory, indicated by a dashed line known in the trade as a 1004 (4m gap, 2 m line). The other is a mandatory cycle lane, indicated by a solid white line. The madatory part refers to cars not being allowed in it rather than cyclists must use it.
However, an advisory lane can be enhanced with a restriction along the kerb, normally a single or double yellow line outside London.
From the sounds of it, he can park there; it’s just not very considerate to other road users.
However, if a police officer considers that the driver is causing an obstruction, he can issue for that whether or not there are road markings there.
HTH