LTM member in pool has status offline (disabled by parent)

Administratively disabled means that someone disabled the node on the F5 directly (on the GUI / CLI / programmatically).

When looking at the pool member in the F5 configuration utility (GUI):

• The diamond shape indicates that its health check is failing
• The grey colour indicates that it was disabled by parent (i.e. at a node level)

The reason why you are seeing a grey diamond symbol as opposed to a black diamond symbol is due to the fact that it was disabled at a node (parent) level.

If it was disabled at a pool member level, then you would have seen a black diamond symbol instead.

'Disabled by parent' on a pool member means that the node (the parent) has been administratively disabled (session user-disabled):

ltm node 1.2.3.4 {
    address 1.2.3.4
    session user-disabled
}

Ltm::Pool: POOL-TEST
  Availability           : offline
  State                  : enabled
  Reason                 : The children pool member(s) are down
  Available Members      : 0
  | Ltm::Pool Member: 1.2.3.4:80
  |   Availability   : offline
  |   State          : disabled-by-parent
  |   Reason         : Pool member has been marked down by a monitor

This is greatly represented in the network map tool, I suggest you take a look at that as well! 
Object tree view is extremely clear and you can easily check dependencies.