No. The state pension in Estonia is regulated by the State Pension Insurance Act. By law, state pension payments are received by:
permanent residents of Estonia;
aliens residing in Estonia on the basis of temporary residence permits or temporary right of residence.
In Estonia, old-age pensions are paid to people aged 65 and over. However, if you don't actually live in Estonia on a permanent basis, nor make social contributions for more than 15 years, you won't receive the state pension regardless of age.
Essentially, while your Estonian company has to pay social tax on your board member salary there are no tangible benefits to gain (no state pension or health insurance in Estonia). This does make the social tax payments seem useless, but unfortunately they are part of the package which comes with the Estonian legal environment.