Why is Promethium the only radioactive Lanthanide?
Lanthanoids are not generally supposed to be radioactive, with exception of mentioned very long lived radioisotopes. 

The longest half-life radioactive nuclides of lanthanoids ( in years )


  $\ce{^{150}Nd}:\ \pu{6.7e18}$
  $\ce{^{151}Eu}:\  \pu{5e18}$
  $\ce{^{148}Sm}:\  \pu{7e15}$
  $\ce{^{144}Nd}:\   \pu{2.29e15}$
  $\ce{^{152}Gd}:\  \pu{ 108e12}$
  $\ce{^{147}Sm}:\ \pu{106e9}$
  $\ce{^{138}La}:\ \pu{ 102e9}$
  $\ce{^{176}Lu}:\ \pu{ 38.5e9}$   


Promethium is rather a victim of circumstances there was no free position to have at least one stable isotope. It is related also to the Liquid drop model.

There are few semi-empirical rules, related to proton and neutron fermionic pairing within the nucleus model:

Elements with the odd proton number have maximally 2 stable isotopes.

2 Isotopes of adjacent elements with the same nucleon number, like $\ce{^N_{A}X, ^N_{A+1}Y}$, are not both stable. ( But some of these unstable ones  have very long half-life as seen above.)

Isotopes not close enough to the Valley of stability are not stable, as they have too little or too many neutrons.

The stability of isotopes of similar nucleon number generally decreases in order 


both proton/neutron numbers are even
one number is odd, the other is even.
both numbers are odd. 


For Pm and Tc, both have odd proton numbers what gives them disadvantage.

All their possible stable isotope candidates are already taken by their  neighbors as energetically preferred variants. Therefore all isotopes of given 2 elements beta decay to nuclei of adjacent elements with less energy.