Kent Online Parish Clerks
The Answer - "Pd." versus "p"
The definitive authority that permits a fuller explanation of the abbreviations "Pd" and "P" as encountered in the church registers from late-1783 can be discerned among the provisions of An Act for granting to his Majesty a Stamp-duty on the Registry of Burials, Marriages, Births, and Christenings. 
The leading preamble of the Act sets out the date of inception of the duty, i.e. 1 October 1783, the amount of the duty and the items which were to be charged with the duty, vizt.
...be it enacted by the King's most Excellent Majesty, by and with the Advice and Consent of the Lords Spiritual and Temporal, and Commons in this present Parliament assembled, and by the Authority of the same, That, from and after the first Day of October, one thousand seven hundred and eighty-three, there shall be charged, levied, and paid unto And for the Use of his Majesty, his Heirs and Successors, the new Duty following ; (that is to say,) Upon the Entry of any Burial, Marriage, Birth, or Christening, in the Register of any Parish, Precinct, or Place in Great Britain, Stamp-duty of three Pence.
Next, section VI of that Act charges every incumbent with the responsibility for recording in his registers of BMB the fact that the tax had been collected unless he had obtained a license for exemption from this provision of the Act. Previous sections charged every incumbent with the responsibility for collecting the appropriate amount of duty per event and permitting him to retain 10% of the total collected as a commission for his pains in so doing.
The practical application of the requirements of section VI of the Act to the everyday workings of the clergy seems to have been to mark the entries with some form of notation acknowledging that payment had been received. Hence, many registers - and I have examined close to 300 throughout east Kent - contain a simple notation of 'paid', 'tax paid', 'pd'. Often simply the paupers are noted with a letter 'p' inscribed next to the respective entry.
Section VI reads in part
VI. Provided always, and be it further enacted, That no Parson, Vicar, Curate, or other Person, shall be subject to any the Penalties or Forfeitures in this Act mentioned, for entering, or causing to be entered, any Burial, Marriage, Birth, or Christening, in any Parish Register, without any Marks or Stamps thereon, where a Licence, under the Hands of three of the Commissioners for the Time being appointed to put this Act in Execution, or some Officer or Officers by them impowered, shall have been granted, signifying their or his Leave or Approbation that the Entry of any Burial, Marriage, Birth, or Christening, to be written in such Register, may be therein written without any Marks or Stamps thereon...
Section VII of the Act prohibits the clergy from collecting the tax from certain of the poor such as those belonging to a workhouse or hospital or whose parents are in receipt of parish poor-relief funds. Again, this section implies that the incumbent would have to have some method by which he, or any other person, would be able to identify those events that were performed for the poor. Herein lies the origin of the 'P' notation found in many registers contemporary with the tax, the 'P' indicating a pauper or other individual exempted from the tax.
VII. Provided also, and be it further enacted, That nothing herein contained shall extend to charge the Entry in any Parish Register of the Burial of any Person who shall be buried from any Workhouse or Hospital, or at the sole Expence of any Charity ; nor the Entry in any Parish Register of the Birth or Christening of any Child whose Parents shall receive, at the Time of the Birth and Christening of such Child, any Parish Relief.
A liberal interpretation of who was qualified for exemption from payment for the tax on a baptism was often adopted by the incumbent of a parish on the strength of the words in the section above, "...time of birth and christening...any Parish Relief".
Based on my experience with Overseers Accounts, not only in relation to my own family but to others, as well, that it would be made relatively clear from those accounts if a family qualified for exemption based upon being in receipt of "any Parish Relief", which could range from a simple payment for clothes or shoes to weekly payments of poor relief. In any one case, if suspicion exists as to the economic status of a family, I would strongly urge the examination of the Overseer's Accounts for records of payments made to or on behalf of that family.
Generally, I have noticed a high incidence of 'P' descriptions attached to baptisms and burials in the registers of some parishes. Other registers are unusually lacking in what one would consider to be a reflection of an average occurrence of poor within a parish. Without having examined the Overseer's Accounts of every parish where anomalies have been found, I can only think that perhaps the high incident rate of baptisms and burials exempt from the tax were performed by overtly sympathetic clerics. It seems that many clerics were highly sensitive and empathetic to the stringent economic realities endured by the members of their congregation and did what they could, within the confines of secular and ecclesiastical law, to relieve the expense of unpopular taxes.
Thus, we have a very long-winded, but definitive answer to the "Pd" and "P" notations appearing in the church registers from late-1783. The Act was repealed 10 years later. It should also be noted that Quakers, by virtue of this same Act, were liable to the same levy and were likewise required to keep a proper register of payments received.
Several interesting sociological events occurred centered around this Act. Immediately prior to its enactment there was a bulge in christenings with a fair number of those christenings being of older children. Immediately following the repeal of the Act a similar bulge of christenings occurred, again including those of older children. Many christenings and some marriages were simply 'put off' until a better time. Many people just did not have the additional funds to have their routinely arriving children christened consistently while similarly less financially stable couples were often unable to afford the added tax that would be due upon their marriage. The one event that not very many escaped, however, was the burial of family members. But, again, this Act also created a situation whereby family members would not come forward to pay for a burial forcing the parish to bury the deceased at its expense.
To summarize the answer to the question, then, a simple notation of 'paid', 'tax paid', 'pd' in a register likely means just that, that the tax had been paid. A simple notation of 'p' in a register likely implies that the individual subject of the event was poor and no tax was payable.